Reflection theorems for number rings
Abstract
The Ohno-Nakagawa reflection theorem is an unexpectedly simple identity relating the number of -classes of binary cubic forms (equivalently, cubic rings) of two different discriminants , ; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we provide a framework for generalizing this theorem using a global and local step. The global step uses Fourier analysis on the adelic cohomology of a finite Galois module, modeled after the celebrated Fourier analysis on used in Tate’s thesis. The local step is combinatorial, more elementary but much more mysterious. We establish reflection theorems for binary quadratic forms over number fields of class number , and for cubic and quartic rings over arbitrary number fields, as well as binary quartic forms over ; the quartic results are conditional on some computational algebraic identities that are probabilistically true. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field.
Part I Introduction
1 Introduction
1.1 Historical background
In 1932, using the then-new machinery of class field theory, Scholz [ScholzRefl] proved that the class groups of the quadratic fields and , whose discriminants are in the ratio , have -ranks differing by at most . This is a remarkable early example of a reflection theorem. A generalization due to Leopoldt [Leopoldt] relates different components of the -torsion of the class group of a number field containing when decomposed under the Galois group of that field. Applications of such reflection theorems are far-ranging: for instance, Ellenberg and Venkatesh [EV] use reflection theorems of Scholz type to prove upper bounds on -torsion in class groups of number fields, while Mihăilescu [MihCat2] uses Leopoldt’s generalization to simplify a step of his monumental proof of the Catalan conjecture that and are the only consecutive perfect powers. Through the years, numerous reflection principles for different generalizations of ideal class groups have come into print. A very general reflection theorem for Arakelov class groups is due by Gras [Gras].
A quite different direction of generalization was discovered by accident in 1997: The following relation was conjectured by Ohno [Ohno] on the basis of numerical data and proved by Nakagawa [Nakagawa], for which reason we will call it the Ohno-Nakagawa (O-N) reflection theorem:
Theorem 1.1 (Ohno–Nakagawa).
For a nonzero integer , let be the number of -orbits of binary cubic forms
of discriminant , each orbit weighted by the reciprocal of its number of symmetries (i.e. stabilizer in ). Let be the number of such orbits such that the middle two coefficients are multiples of , weighted in the same way.
Then for every nonzero integer , we have the exact identity
| (1) |
By the well-known index-form parametrization (see 6.9 below), also counts the cubic rings of discriminant over , weighted by the reciprocal of the order of the automorphism group. It turns out that counts those rings for which for every . When is a fundamental discriminant, the corresponding cubic extensions are closely related, via class field theory, to the -class group of and we get back Scholz’s reflection theorem, as Nakagawa points out ([Nakagawa], Remark 0.9).
Theorem 1.1 was quite unexpected, because -orbits of binary cubics have been tabulated since Eisenstein without unearthing any striking patterns. Even the exact normalizations , had been in use for over two decades. They appear in the Shintani zeta functions
a family of Dirichlet series which play a prominent role in understanding the distribution of cubic number fields, similar to how the famous Riemann zeta function controls the distribution of primes. As Shintani proved as early as 1972 [Shintani], the Shintani zeta functions satisfy a matrix functional equation (see Nakagawa [Nakagawa], eq. (0.1))
| (2) |
The condition that divide and is equivalent to requiring that the cubic form is integer-matrix, that is, its corresponding symmetric trilinear form
has integer entries. This condition arose in Shintani’s work by taking the dual lattice to under the pairing
| (3) |
which plays a central role in proving the functional equation. However, as we will find, the pairing (3) does not figure in the proof of our reflection theorems, which indeed often relate lattices that are not dual under it.
Using the functional equation, Shintani proved that the admit meromorphic continuations to the complex plane with simple poles at and , inspiring him to conjecture that the number of cubic fields of positive or negative discriminant up to has the shape
for suitable constants and . This conjecture was proven by Bhargava, Shankar, and Tsimerman [BST-2ndOrd] and independently by Taniguchi and Thorne [TT_rc]. Neither proof needs the Ohno-Nakagawa reflection theorem (Theorem 1.1), which appears in the notation of Shintani zeta functions in the succinct form
| (4) |
Remark 1.2.
In the earlier papers, the term “Ohno-Nakagawa identities” was used, referring to the pair (4). Our work confirms the intuition that, despite the different scalings, both identities are essentially one theorem.
1.2 Methods
Several proofs of O-N are now in print ([Nakagawa, Marinescu, OOnARemarkable, Gao]), all of which consist of two main steps:
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•
A “global” step that uses global class field theory to understand cubic fields, equivalently -orbits of cubic forms;
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A “local” step to count the rings in each cubic field, equivalently the -orbits in each -orbit, and put the result in a usable form.
In this paper, the distinction between these steps will be formalized and clarified.
For the global step, we take inspiration from Tate’s celebrated thesis [Tate_thesis], which uses Fourier analysis on the adeles to give illuminating new proofs of the functional equations for the Riemann -function and various -functions. Taniguchi and Thorne (see [TT_oexp]) used Fourier analysis on the space of binary cubic forms over to get the functional equation for the Shintani zeta function of forms satisfying local conditions at primes. Despite the similarities, their work is essentially independent from ours. We are also inspired by a remark due to Calegari in a paper of Cohen, Rubinstein-Salzedo, and Thorne ([CohON], Remark 1.6), pointing out that their reflection theorem counting dihedral fields of prime order can also be derived from a theorem of Greenberg and Wiles for the sizes of Selmer groups in Galois cohomology.
We present a notion of composed variety, a scheme over the ring of integers of a number field admitting an action of an algebraic group over . Our guiding example is the scheme of binary cubic forms of discriminant with its action of . The term “composed” refers to the presence of a composition law on the orbits, which relate naturally to a Galois cohomology group . Our (global) reflection theorems can be stated as saying that two composed varieties , have the same number of -points, with a suitable weighting. Introducing a new technique of Fourier analysis on the adelic cohomology group , based on Poitou-Tate duality, we present a generalized reflection engine (Theorems 8.12 and LABEL:thm:main_compose_multi) that reduces global reflection theorems to local reflection theorems, that is, statements involving only the -points of and for a single place of . A typical case is Theorem LABEL:thm:O-N_cubic_local.
These local reflection theorems are approachable by elementary methods but can be difficult to prove. We present two kinds of proofs. The first is a bijective argument involving Bhargava’s self-balanced ideals that is very clean but has only been discovered at the “tame primes” ( in the cubic case, in the quartic). The second is by explicitly computing the number of orders of given resolvent in a cubic or quartic algebra. We express it as a generating function in a number of variables depending on the splitting type of the resolvent. The generating function is rational, and local reflection can be written as an equality between two rational functions; but these functions are so complicated that the best approximation to a proof of the identity that we can find is a Monte Carlo proof, namely, substituting random values for the variables in some large finite field and verifying that the equality holds. The reader is invited to recheck this verification using the source code in Sage that will be made available with the final version of this paper.
1.3 Results
We are able to prove O-N for binary cubic forms over all number fields , verifying and extending the conjectures of Dioses [Dioses, Conjecture 1.1]. However, we go further and ask whether every -invariant lattice within the space of binary cubic forms admits an O-N-style reflection theorem. Over , this question was answered affirmatively for each of the ten invariant lattices by Ohno and Taniguchi [10lat]. Over , such lattices were classified by Osborne [Osborne], and they differ from one another only at the primes dividing and . The lattices at yield an elegant reflection theorem (Theorem LABEL:thm:O-N_traced) in which the condition , where is an ideal dividing in , reflects to , the complementary divisor. At , the corresponding reflection theorems still exist, though they become difficult to write explicitly: see Theorem LABEL:thm:invarlat.
We also find a new reflection theorem (Theorem LABEL:thm:O-N_quad) counting binary quadratic forms, not by discriminant, but by a curious invariant: the product of the discriminant and the leading coefficient. Over , the reflection theorem (Theorem LABEL:thm:O-N_quad_Z) has the potential to be proved simply using quadratic reciprocity, eschewing the machinery of Galois cohomology, though it seems unlikely that the theorem would have ever been discovered without it.
Nakagawa has also conjectured [NakPairs] a reflection theorem for pairs of ternary quadratic forms, which parametrize quartic rings. The natural invariant to count by is the discriminant, but it is more natural from our perspective to subdivide further and ask for a reflection theorem for rings with fixed cubic resolvent, which holds in the known cases [NakPairs, Theorem 1]. Here our global framework applies without change, but the local enumeration of orders in a quartic field presents formidable combinatorial difficulties, especially in the wildly ramified (-adic) setting, which have been attacked in another work of Nakagawa [NakOrders]. Our methods have the potential to finish this work, but because we count by resolvent rather than discriminant, our answers do not directly match his.
The process of proving local quartic O-N leads us down some fruitful routes that do not at first sight have any connection to reflection theorems or to the enumeration of quartic rings. These include new cases of the Igusa zeta functions of conics (Lemmas LABEL:lem:conic_1 and LABEL:lem:conic_pi) and a result on the average value of a quadratic character on a box in a local field (Theorem LABEL:thm:char_box). If quartic O-N holds true in all cases, it implies that the cubic resolvent ring (in the sense of Bhargava) of a maximal quartic order has a second natural characterization: it is the “conductor ring” for which the Galois-naturally attached extension is a ring class field (Theorem* LABEL:thm*:cond_ring).
1.4 Outline of the paper
In Section 2, we state and give examples of the main global reflection theorems of the paper over , in a fashion that requires a minimum of prior knowledge, for the end of further diffusing interest in, and appreciation of, the beauty of number theory.
In Part II, we lay out preliminary matter, much of which is closely related to results that have appeared in the literature but under different guises. It includes a simple characterization (Proposition 4.21) of Galois in terms of étale algebras whose Galois group is a semidirect product. It also includes a theorem (Theorem 7.1) on the structure of in the case that is local and (with any Galois structure), which will be invaluable in what follows.
In Part III, we lay out the framework of composed varieties, on which we perform the novel technique of Fourier analysis of the local and global Tate pairings to get our main local-to-global reflection engine (Theorems 8.12 and LABEL:thm:main_compose_multi). The remainder of the paper will concern applications of this engine.
In Part LABEL:part:first, we prove two relatively simple reflection theorems: one for quadratic forms (Theorem LABEL:thm:O-N_quad), and a version of the Scholz reflection principle for class groups of quadratic orders (Theorem LABEL:thm:Scholz_for_locally_dual_orders).
In Part LABEL:part:cubic, we prove our extensions of Ohno-Nakagawa for cubic forms and rings.
The quartic case is dealt with in Parts LABEL:part:quartic and LABEL:part:quartic_count: the first part dealing with the bijective methods, and the second with the (long) work of explicitly counting orders in each quartic algebra. The case of partially ramified cubic resolvent (splitting type ) is still in progress, so we restrict our attention to the four tamely splitting types in the present version.
We conclude the paper with some unanswered questions engendered by this research.
1.5 Acknowledgements
For fruitful discussions, I would like to thank (in no particular order): Manjul Bhargava, Xiaoheng Jerry Wang, Fabian Gundlach, Levent Alpöge, Melanie Matchett Wood, Kiran Kedlaya, Alina Bucur, Benedict Gross, Sameera Vemulapalli, Brandon Alberts, Peter Sarnak, and Jack Thorne.
2 Examples for the lay reader
Fortunately for the non-specialist reader, the statements (though not the proofs) of the main results in this thesis can be stated in a way requiring little more than high-school algebra. We here present these statements and some examples to illustrate them.
2.1 Reflection for quadratic equations
Definition 2.1.
Let be a quadratic polynomial, where the coefficients , , are integers. The superdiscriminant of is the product
of the leading coefficient with the usual discriminant.
Lemma 2.2.
If we replace by in a quadratic polynomial , where is a fixed integer, then the superdiscriminant does not change.
Proof.
This can be verified by brute-force calculation, but the following method is more illuminating. The discriminant is classically related to the two roots of ,
through their difference:
If we replace by , then does not change, and both roots are decreased by , so their difference is unchanged. Therefore is unchanged. ∎
Definition 2.3.
Call two quadratics , equivalent if they are related by a translation . If is a nonzero integer, let be the number of quadratics of superdiscriminant , up to equivalence. Let , , be the number of such quadratics that satisfy certain added conditions:
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For , we require that the middle coefficient be even.
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For , we require that the roots be real, that is, that .
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For , we impose both of the last two conditions.
We are now ready to state a quadratic reflection theorem, the main result of this section.
Theorem 2.4 (“Quadratic O-N”).
For every nonzero integer ,
Proof.
The proof is not easy. See Theorem LABEL:thm:O-N_quad_Z. ∎
It’s not hard to compute all quadratics of a fixed superdiscriminant . The leading coefficient must be a divisor of (possibly negative), and there are only finitely many of these. Then, by replacing by where is an integer nearest to , we can assume that lies in the window . We can try each of the integer values in this window, checking whether
comes out to an integer.
Example 2.5.
There are five quadratics of superdiscriminant :
|
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You might think we left out , but it is equivalent to another quadratic on the list:
So we get the totals
There are quadratics of superdiscriminant :
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Counting carefully, we get
The equalities
are instances of Theorem 2.4. From the same theorem, we derive, without computation, that
This short investigation raises many questions. The superdiscriminant does not seem to have been considered before. Is there an explicit formula for ? Is there an elementary proof of Theorem 2.4? See Example LABEL:ex:QR for a connection to Gauss’s celebrated law of quadratic reciprocity.
2.2 Reflection for cubic equations
Definition 2.6.
For a cubic polynomial
we define the discriminant to be
| (5) |
where are the roots. Explicitly,
| (6) |
There are many transformations of a cubic polynomial that don’t change the discriminant. One is changing to , where is a constant. Another is reversing the coefficients,
Both of these are special cases of the following construction.
Definition 2.7.
Two cubic polynomials , with integer coefficients are equivalent if there is a matrix
whose determinant is such that
A matrix that makes equivalent to itself, that is,
is called a symmetry of . The number of symmetries of is denoted by .
Definition 2.8.
If is a nonzero integer, define to be the number of cubic polynomials
of discriminant , up to equivalence, each counted not once but times, where is the number of symmetries. Define to be the number of cubics of discriminant for which the middle two coefficients, and , are multiples of , up to equivalence, each counted times as before.
We can now state the Ohno-Nakagawa reflection theorem that got this research project started:
Theorem 2.9 (Ohno-Nakagawa; Theorem 1.1).
For every nonzero integer ,
Proof.
Several proofs are in print (see the Introduction). In this paper, we prove this theorem as a special case of Theorem LABEL:thm:O-N_traced. ∎
Example 2.10.
Take . There is just one cubic with integer coefficients and discriminant , namely
The reader may balk at considering a quadratic polynomial as a “cubic” with leading coefficient , but the polynomial can be replaced by any number of equivalent forms, for instance
We will suppress this detail in subsequent examples.
(A program for computing all cubics of a given discriminant is found in the attached file cubics.sage, based on an algorithm of Cremona [Crem_Redn, Crem_Redn_2]). The cubic has six symmetries, which is related to the fact that three linear factors can be permuted in ways. In terms of , the symmetries are
So .
Correspondingly, we look at cubics of discriminant . There are two:
Each admits two symmetries: the first has
and the second has
So and . In particular,
in conformity with Theorem 2.9.
2.3 Reflection for boxes
Bhargava [B3] studied boxes as a visual representation for quartic rings, as cubic polynomials do for cubic rings. We think that reflection holds not only for boxes but for , , and so on. We nearly prove the case in this paper. We are quite far from proving it for the larger boxes.
Definition 2.11.
A box is a pair of integer symmetric matrices. The resolvent of a box is the polynomial
It is a polynomial in , of degree at most . If is the identity matrix, the resolvent devolves into the standard characteristic polynomial.
Definition 2.12.
Two boxes and are equivalent if there is an integer matrix , whose inverse also has integer entries, such that
If are the same pair, then is called a symmetry of . The number of symmetries of will be denoted by .
Conjecture 2.13 (“O-N for boxes”).
Let be a positive odd integer. Let be a polynomial of degree with no multiple roots and only one real root. Denote by the number of boxes with resolvent , up to equivalence, each box weighted by the reciprocal of its number of symmetries. Denote by the number of such boxes with even numbers along the main diagonals of and , weighted the same way. Then
| (7) |
Remark 2.14.
The condition that have no multiple roots (even complex ones) is needed to ensure that there are only finitely many boxes with as a resolvent. The condition that have no more than one real root can be eliminated, but then we must impose conditions on the real behavior of the boxes that are difficult to state succinctly.
Example 2.15.
Take as resolvent , the simplest irreducible cubic. It has one real root and discriminant . There are two boxes with resolvent , up to equivalence:
(These were computed from the balanced pairs and in the number field corresponding to .) Neither has any symmetries besides the two trivial ones, the identity matrix and its negative, so
There are many boxes with resolvent , but just one with even numbers all along the main diagonals of and , namely
(This was computed from the unique quartic ring with resolvent .) It too has only the trivial symmetries, to , in accord with Conjecture 2.13.
2.4 Reflection for quartic equations
There are also reflection theorems that appear when counting quartic polynomials.
Definition 2.16.
If
is a quartic polynomial with integer coefficients, its resolvent is
| (8) |
equivalently, if
then
Remark 2.17.
Cubic resolvents of this type have been used since the 16th century as a step in solving quartic equations. For instance, it is well known that if factors as the product of two quadratics with integer coefficients, then has a rational root (the converse is not true).
Analogously to Definition 2.7, we put:
Definition 2.18.
Two quartic polynomials , with integer coefficients are equivalent if there is a matrix
whose determinant is such that
A matrix that makes equivalent to itself, that is,
is called a symmetry of . The number of symmetries of is denoted by .
We have:
Lemma 2.19.
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a
If two quartics , are equivalent, then their resolvents , are related by a translation
for some integer .
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b
A quartic and its resolvent have the same discriminant
Proof.
Exercise. ∎
As before, our reflection theorem will relate general quartics to quartics satisfying certain divisibility relations. Here the relations are quite peculiar:
Definition 2.20.
A quartic polynomial
is called supereven if , , and are multiples of and is a multiple of .
Not every quartic equivalent to a super-even quartic is itself supereven. (For instance, and are equivalent under the flip , but is not supereven.) We therefore make the following definition.
Definition 2.21.
Two quartic polynomials , with integer coefficients are evenly equivalent if there is a matrix
whose determinant is , and r is even, such that
Such a matrix that makes equivalent to itself, that is,
is called an even symmetry of . The number of even symmetries of is denoted by .
Theorem 2.22 (“Quartic O-N”).
Let be an integer cubic with leading coefficient , no multiple roots, and odd discriminant. Denote by the number of quartics whose resolvent is for some , up to equivalence and weighted by the reciprocal of the number of symmetries. Denote by the number of supereven quartics whose resolvent is for some , up to even equivalence and weighted by the reciprocal of the number of even symmetries. Define by
Then:
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If has one real root, then
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If has three real roots, then we subdivide
where the respective terms count only quartic functions that are always positive, always negative, and have four real roots. We subdivide
Then:
Also, denote by the number of integral symmetric matrices of characteristic polynomial . Then
Proof.
See Theorem LABEL:thm:BQ. ∎
Remark 2.23.
We think that the hypothesis of odd discriminant is removable, but we have not yet finished the proof.
Example 2.24.
Let . By techniques presented in Section LABEL:sec:bq, it is possible to transform the boxes found in example 2.15 into binary quartic forms. We find that there is only one quartic with resolvent , namely
(which, as before, can be transformed by an equivalence to one with nonzero leading coefficient); and four supereven binary quartics with resolvent , namely
All these have one pair of complex roots (as must occur for a resolvent with negative discriminant) and only the trivial symmetries , so
in accord with the first part of the theorem.
Example 2.25.
Consider , a cubic with three real roots. The quartics with resolvent are
which has four real roots, and
which has no real roots and is positive for all real . Each has only the trivial symmetries, so
(Note the discrepancy between and .) Correspondingly, there are eight supereven binary quartics with resolvent :
Thus
This is in accord with the theorem, from which we also learn that
so is not the characteristic polynomial of any integer symmetric matrix, despite having three real roots (which is a necessary, but not a sufficient, condition).
Example 2.26.
Let . Knowing that is the only quartic with cubic resolvent , and it has four symmetries, the powers of , we get
So there are six symmetric matrices with characteristic polynomial . Indeed, they are the diagonal matrices with , , and along the diagonal in any of the possible orders.
3 Notation
The following conventions will be observed in the remainder of the paper.
We denote by and , respectively, the sets of nonnegative and of positive integers.
If is a statement, then
If is a set, then denotes the characteristic function .
An algebra will always be commutative and of finite rank over a field, while a ring or order will be a finite-dimensional, torsion-free ring over a Dedekind domain, containing . An order need not be a domain.
If are elements of a local or global field, a separable closure thereof, or a finite product of the preceding, we write to mean that for some in the appropriate ring of integers . If and , we say that and are associates and write . Note that and may be zero-divisors.
If is a finite set, we let denote the set of permutations of ; thus . If , and if , are elements, we say that and are conjugate if there is a bijection between and under which they correspond. Likewise when we say that two subgroups , are conjugate.
We will use the semicolon to separate the coordinates of an element of a product of rings. For instance, in , the nontrivial idempotents are and .
If is a positive integer, then denotes a primitive th root of unity in , while denotes the th root of unity
Throughout the proofs of the local reflection theorems, we will fix a local field , its valuation , its residue field of order , and a uniformizer . The letter will denote the absolute ramification index ( in the quadratic and quartic cases, in the cubic). We let denote the maximal ideal, and likewise be the maximal ideal of the ring of algebraic integers over ; note that is not finitely generated. We also allow to be applied to elements of , the valuation being scaled so that its restriction to has value group . We use the absolute value bars for the corresponding metric, whose normalization will be left undetermined.
If is a local field, an -pixel is a subset of an affine or projective space over defined by requiring the coordinates to lie in specified congruence classes modulo . For instance, in , a -pixel is the whole space, which is subdivided into -many -pixels for each .
If is a finite-dimensional, locally free algebra over a ring, we denote by the subgroup of units of norm . The group operation is implicitly multiplication, so , for instance, denotes the th roots of unity of norm .
Part II Galois cohomology
4 Étale algebras and their Galois groups
4.1 Étale algebras
If is a field, an étale algebra over is a finite-dimensional separable commutative algebra over , or equivalently, a finite product of finite separable extension fields of . A treatment of étale algebras is found in Milne ([MilneFields], chapter 8): here we summarize this theory and prove a few auxiliary results that will be of use.
An étale algebra of rank admits exactly maps (of -algebras) to a fixed separable closure of . We call these the coordinates of ; the set of them will be called or simply . Together, the coordinates define an embedding of into , which we call the Minkowski embedding because it subsumes as a special case the embedding of a degree- number field into , which plays a major role in algebraic number theory, as in Delone-Faddeev [DF].
For any element of the absolute Galois group , the composition with any coordinate is also a coordinate , so we get a homomorphism ) such that
for all . This gives a functor from étale -algebras to -sets (sets with a -action), which is denoted in Milne’s terminology. A functor going the other way, which Milne calls , takes to
| (9) |
Proposition 4.1 ([MilneFields], Theorem 7.29).
The functors and establish a bijection between
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étale extensions of degree , up to isomorphism, and
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-sets of size up to isomorphism; that is to say, homomorphisms , up to conjugation in .
Moreover, the bijection respects base change, in the following way:
Proposition 4.2.
Let be a field extension, not necessarily algebraic, and let be an étale extension of degree . Then is étale over , and the associated Galois representations , are related by the commutative diagram
| (10) |
Proof.
That is étale is standard (see Milne [MilneFields], Prop. 8.10). For the second claim, consider the natural restriction map . It is injective, since a linear map out of is determined by its values on ; and since both sets have the same size, is surjective and is hence an isomorphism of -sets (the -structure on arising by restriction from the -structure). ∎
We will use this proposition most frequently in the case that is a global field and one of its completions. The resulting is then the product of the completions of at the places dividing . Note the departure from the classical habit of studying the completion at each place individually. The preservation of degrees, will be important for our applications.
4.2 The Galois group of an étale algebra
Define the Galois group of an étale algebra to be the image of its associated Galois representation . It transitively permutes the coordinates corresponding to each field factor. For example, if is a quartic field, then is one of the five (up to conjugacy) transitive subgroups of , which (to use the traditional names) are , , , , and . Galois groups in this sense are used in the tables of cubic and quartic fields in Delone-Faddeev [DF] and the Number Field Database [NFDB]. Note that the Galois group is defined whether or not is a Galois extension. If it is, then the Galois group is simply transitive and coincides with the Galois group in the sense of Galois theory.
Important for us will be two notions pertaining to the Galois group.
Definition 4.3.
Let be a subgroup. A -extension of is a degree- étale algebra with a choice of subgroup that is conjugate to and contains , plus a conjugacy class of isomorphisms : the conjugacy being in , not in . The added data is called a -structure on .
Proposition 4.4.
-extensions up to isomorphism are in bijection with homomorphisms , up to conjugation in .
Proof.
Immediate from Proposition 4.1. ∎
Example 4.5.
is a -extension (taking ), indeed its Galois group is isomorphic to ; and admits two distinct -structures, as there are two ways to identify with its image in , which are conjugate in but not in . Likewise, admits six -structures, one for each embedding of into , as its Galois group is trivial.
4.3 Resolvents
This will be an important notion.
Definition 4.6.
Let , be subgroups and be a homomorphism. Then for every -extension , the corresponding may be composed with to yield a map , which defines an étale extension of degree . This is called the resolvent of under the map .
Example 4.7.
Since there is a surjective map , every quartic étale algebra has a cubic resolvent . This resolvent appears in Bhargava [B3], but it is much older than that. It is generated by a formal root of the resolvent cubic that appears when a general quartic equation is to be solved by radicals.
Example 4.8.
Likewise, the sign map can be viewed as a homomorphism , attaching to every étale algebra a quadratic resolvent . If is generated by a polynomial , and if , then it is not hard to see that where is the polynomial discriminant. Note that still exists even if . We have that is split if and only if the Galois group is contained in the alternating group .
Example 4.9.
The dihedral group has an outer automorphism, because rotating a square in the plane by does not preserve the square but does preserve every symmetry of the square. This map associates to each -algebra a new -algebra , not in general isomorphic. This is the classical phenomenon of the mirror field. For instance, if , then
Both and have the same Galois closure, a -octic extension of . Likewise, the outer automorphism of permits the association to each sextic étale algebra a mirror sextic étale algebra .
Example 4.10.
The Cayley embedding is an embedding of any group into , acting by left multiplication. The Cayley embedding attaches to every étale algebra of degree an algebra of degree with an -torsor structure. This is none other than the -closure of , constructed by Bhargava in a quite different way in [B3, Section 2].
More generally, for any , the Cayley embedding allows one to associate to each -extension a -torsor , which we may call the -closure of . The name “closure” is justified by the following observation: if is a transitive subgroup, then, since any transitive -set is a quotient of the simply transitive one, we can embed into by Proposition 4.11 below. More generally, -closures of ring extensions, not necessarily étale or even reduced, have been constructed and studied by Biesel [Biesel_thesis, Biesel].
If is invertible, as in many of the above examples, then the map from -extensions to -extensions is also invertible: we say that the two extensions are mutual resolvents.
4.4 Subextensions and automorphisms
The Galois group holds the answers to various natural questions about an étale algebra. The next two propositions are given without proof, since they follow immediately from the functorial character of the correspondence in Proposition 4.1
Proposition 4.11.
The subextensions of an étale extension , correspond to the equivalence relations on stable under permutation by , under the bijection
Remark 4.12.
Note that if is a Galois field extension, the image of is a simply transitive subgroup , and identifying with , the stable equivalence relations are just right congruences modulo subgroups of : so we recover the Galois correspondence between subgroups and subfields.
The Galois group is not a group of automorphisms of . However, the automorphisms of as a -algebra can be described in terms of the Galois group readily.
Proposition 4.13.
Let be Minkowski-embedded by its coordinates . Then the automorphism group is given by permutations of coordinates,
for in the centralizer of the Galois group.
(For groups, the centralizer of in is the subgroup of elements of that commute with every element of .)
This provides a characterization, in terms of the Galois group, of rings having various kinds of automorphisms.
-
•
Since is abelian, any étale algebra of rank has a unique non-identity automorphism, the conjugation .
-
•
If has rank , automorphisms of of order whose fixed algebra is of rank are in bijection with -structures on . Indeed, the conditions force to correspond to the permutation or one of its conjugates, and the centralizer of this permutation is .
-
•
Particularly relevant is the case that has a complete set of automorphisms that permute the coordinates simply transitively: this is a generalization of a Galois field extension called a torsor. This case is sufficiently important to merit its own subsection.
4.5 Torsors
Definition 4.14.
Let be a finite group. A -torsor over is an étale algebra over equipped with an action of by automorphisms that permute the coordinates simply transitively, that is, such that is isomorphic to
with acting by right multiplication on the indices.
Proposition 4.15.
Let be a group of order . An étale algebra is a -torsor if and only if it is a -extension, where is embedded into by the Cayley embedding ( acting on itself by left multiplication). Moreover, there is a bijection between
-
•
-torsor structures on , up to conjugation in , and
-
•
-structures on .
The bijection is given in the following way: there is a labeling of the coordinates of with the elements of such that the Galois action is by left multiplication
| (11) |
while the torsor action is by right multiplication
| (12) |
Proof.
We first claim that the only elements of commuting with all right multiplications are left multiplications, and vice versa. If is a permutation commuting with left multiplications, then
so is a right multiplication. So the embedded images of in given by left and right multiplication (which are conjugate under the inversion permutation ) are centralizers of one another. It is then clear that conjugates of in that contain are in bijection with conjugates that commute with . This establishes the first assertion. For the bijection of structures, if an embedding is given, then we can label the coordinates with elements of so that acts on them by multiplication; then gets identified with by the corresponding right action. The only ambiguity is in which embedding is labeled with the identity element; if this is changed, one computes that the resulting identification of with is merely conjugated, so the map is well defined. The reverse map is constructed in exactly the same way. ∎
Here is another perspective on torsors.
Proposition 4.16.
-torsors over a field , up to isomorphism, are determined by their field factor, a Galois extension equipped with an embedding up to conjugation in .
Proof.
If is a -torsor, then since permutes the coordinates simply transitively, all the coordinates have the same image; that is, the field factors of are all isomorphic to a Galois extension . The torsor operations fixing one field factor of realize the Galois group as a subgroup of ; changing the field factor and/or the identification corresponds to conjugating the map by an element of .
Conversely, suppose and an embedding
are given. Let be coset representatives for . Then must map any field factor isomorphically onto the remaining field factors , each occurring once. To finish specifying the -action on , it suffices to determine for each . Factor for some , . Then for each , , and the value of this is known because the -action on is known. It is easy to see that we get one and only one consistent -torsor action in this way. ∎
Because all field factors of a torsor are isomorphic, we will sometimes speak of “the” field factor of a torsor.
4.5.1 Torsors over étale algebras
On occasion, we will speak of a -torsor over , where is itself a product of fields. By this we simply mean a product where each is a -torsor over . This case is without conceptual difficulty, and some theorems on torsors will be found to extend readily to it, such as the following variant of the fundamental theorem of Galois theory:
Theorem 4.17.
Let be a -torsor over an étale algebra . For each subgroup ,
-
(a)
The fixed algebra is uniformly of degree over (that is, of this same degree over each field factor of );
-
(b)
is an -torsor over , under the same action;
-
(c)
If is normal, then is also a -torsor over , under the natural action.
Proof.
Adapt the relevant results from Galois theory. ∎
4.6 A fresh look at Galois cohomology
Galois cohomology is one of the basic tools in the development of class field theory. It is usually presented in a highly abstract fashion, but certain Galois cohomology groups, specifically for finite , have explicit meaning in terms of field extensions of . It seems that this interpretation is well known but has not yet been written down fully, a gap that we fill in here. We begin by describing Galois modules.
Proposition 4.18 (a description of Galois modules).
Let be a finite abelian group, and let be a field. Let denote the subset of elements of of maximal order , the exponent of . The following objects are in bijection:
-
a
Galois module structures on over , that is, continuous homomorphisms ;
-
b
-torsors ;
-
c
-extensions , where in the natural way;
-
d
-extensions , where in the natural way.
Proof.
For item d to make sense, we need that generates ; this follows easily from the classification of finite abelian groups.
We will denote with its Galois-module structure coming from these bijections by , , or . Note that , , and are mutual resolvents.
Example 4.19.
For example (and we will return to this case frequently), if we let be the smallest group with nontrivial automorphism group: . Then the Galois module structures on are in natural bijection with -torsors over , that is, quadratic étale extensions . If , these can be parametrized by Kummer theory as , . The value corresponds to the split algebra and to the module with trivial action. We have an isomorphism
of -sets, and of Galois modules if the right-hand side is given the appropriate group structure with as identity.
In particular, the Galois-module structures on form a group : the group operation can also be viewed as tensor product of one-dimensional -vector spaces with Galois action.
4.6.1 Galois cohomology
Note that the zeroth cohomology group has a ready parametrization:
Proposition 4.20.
Let be a Galois module. The elements of are in bijection with the degree- field factors of .
Proof.
Proposition 4.18 establishes an isomorphism of -sets between the coordinates of and the points of . A degree- field factor corresponds to an orbit of on of size , which corresponds exactly to a fixed point of on . ∎
Deeper and more useful is a description of . For an abelian group , let be the semidirect product under the natural action of on . We can describe more explicitly as the group of affine-linear transformations of ; that is, maps
composed of an automorphism and a translation, the group operation being composition. In particular, we have an embedding
Proposition 4.21 (a description of ).
Let be a Galois module.
-
a
is in natural bijection with the set of continuous homomorphisms such that the following triangle commutes:
(13) -
b
is in natural bijection with the set of such up to conjugation by .
-
c
is also in natural bijection with the set of -extensions (with respect to the embedding ) equipped with an isomorphism from their resolvent -torsor to .
Proof.
By the standard construction of group cohomology, is the group of continuous crossed homomorphisms
Send each to the map
It is easy to see that the conditions for to be a homomorphism are exactly those for to be a crossed homomorphism, establishing a. For b, we observe that adding a coboundary to a crossed homomorphism is equivalent to post-conjugating the associated map by . As to c, a -extension carries the same information as a map up to conjugation by the whole of . Specifying the isomorphism from the resolvent -torsor to means that the map is known exactly, not just up to conjugation. Hence is known up to conjugation by . ∎
Remark 4.22.
The zero cohomology class corresponds to the extension , with its structure given by the embedding . This can be seen to be the unique cohomology class whose corresponding -extension has a field factor of degree .
If is a local field, a cohomology class is called unramified if it is represented by a cocycle that factors through the unramified Galois group . The subgroup of unramified coclasses is denoted by . If itself is unramified (and we will never have to think about unramified cohomology in any other case), this is equivalent to the associated étale algebra being unramified.
If is a Galois module and is the Galois module corresponding to a -extension , we can also take the -closure of , a -torsor which fits into the following diagram:
| (14) |
Because of the semidirect product structure of , we have . It is also worth tabulating the permutation representations of finite groups that yield each of the étale algebras discussed here:
| (15) |
4.6.2 The Tate dual
If is a Galois module and the exponent of is not divisible by , then
is also a Galois module, called the Tate dual of . The modules and have the same order and are isomorphic as abstract groups, though not canonically; as Galois modules, they are frequently not isomorphic at all.
Example 4.23.
If is one of the order- modules studied in Example 4.19, then the relevant is
Examining the Galois actions (here it helps to use the theory of -sets of size presented in Knus and Tignol [QuarticExercises]), we see that
This explains the pattern in the Scholz reflection theorem and its generalizations, including cubic Ohno-Nakagawa.
Example 4.24.
A module of underlying group is always self-dual, regardless of what Galois-module structure is placed on it. This can be proved by noting that has a unique alternating bilinear form
Being unique, it is Galois-stable and induces an isomorphism .
Particularly notable for us are the cases when is the full symmetric group , for then every étale algebra of degree has a (unique) -affine structure. It is easy to see that there are only four such cases:
-
•
,
-
•
,
-
•
,
-
•
, .
For degree exceeding , not every étale algebra arises from Galois cohomology, a restriction that plays out in the existing literature on reflection theorems. For instance, Cohen, Rubinstein-Salzedo, and Thorne [CohON] prove a reflection theorem in which one side counts -dihedral fields of prime degree . From our perspective, these correspond to cohomology classes of an whose Galois action is by . The Tate dual of such an can have Galois action by the full , and indeed they count extensions of Galois group on the other side of the reflection theorem. This will appear inevitable in light of the motivations elucidated in Part III.
5 Extensions of Kummer theory to explicitize Galois cohomology
Now that Galois cohomology groups have been parametrized by étale algebras, can invoke parametrizations of étale algebras by even more explicit objects. The most familiar instance of this is Kummer theory, an isomorphism
coming from the long exact sequence associated to the Kummer sequence
In favorable cases, the cohomology of other Galois modules can be embedded into for some finite extension of .
We first state the hypothesis we need:
Definition 5.1.
Let be a finite Galois module of exponent over a field , and let be a Galois-stable generating set of . We say that equipped with is a good module if the natural map of Galois modules
is split, that is, its kernel admits a Galois-stable complementary direct summand . Such a direct summand is known as a good structure on .
Proposition 5.2.
The following examples of a Galois module with generating set are good:
-
a
, with any action, and .
-
b
, with any action preserving a basis .
-
c
, with , with an action that preserves a hyperbasis , that is, a generating set of elements with sum .
Proof.
-
a
Here the Galois modules are representations of over . Since the group and field are of coprime order, complete reducibility holds: any subrepresentation is a direct summand. In fact, is the regular representation, is the tautological representation in which each acts by multiplication by , and can be taken (uniquely in general) to be the product of all the other isotypical components of .
-
b
Here the natural map is an isomorphism, so .
-
c
Here the natural map is the quotient by the one-dimensional space
This space has a Galois-stable direct complement, namely the kernel of the linear functional
Proposition 5.3.
Let be a Galois module with a good structure , and let be the resolvent algebra corresponding to the -set . For any Galois module with underlying group , there is a natural injection
as a direct summand. The cokernel is naturally isomorphic to
Proof.
We use the good structure
to embed
Since is a direct summand, this is an injection with cokernel naturally isomorphic to . It remains to construct an isomorphism
If decomposes as a product
of field factors corresponding to the orbits of on , then has a corresponding decomposition
where is none other than the induced module . Its cohomology is computed by Shapiro’s lemma:
This is the desired isomorphism. ∎
We can harness Kummer theory to parametrize cohomology of other modules as follows.
Theorem 5.4 (an extension of Kummer theory).
Let be a finite Galois module, and assume that is not divisible by . Let act on the set of surjective characters through its actions on and , and let be the étale algebra corresponding to this -set.
-
a
There is a natural group homomorphism
-
b
If is cyclic of prime order, then is injective, is naturally a -torsor, and
If , then the image simplifies to
and the -extension corresponding to a given of norm can be described as follows: Define a -linear map
where is chosen to have norm , and ranges through the set
of cube roots of in of norm . Then
-
c
If , then is injective and
Moreover, the -extension corresponding to a given of norm can be described as follows: Define a -linear map
where is chosen to have norm , and ranges through the set
of square roots of in of norm . Then
Proof.
If is a surjective character, let be the fixed field of the stabilizer of ; thus is the field factor of corresponding to the -orbit of . If are orbit representatives, we can map
This yields our map . Alternatively, note that by Shapiro’s lemma,
where
a Galois module under the action
Under this identification, it is not hard to check that , where is the inclusion given by
Although is injective (because the characters of maximal order generate the group of all characters), it is not obvious whether induces an injection on cohomology, nor what the image is. What makes the modules in parts b and c tractable is that, in these cases, is a good generating set for , so is a direct summand of . In part b, we can identify
as a twist of the regular representation of over . Since has a complete set of st roots of unity, this representation splits completely into one-dimensional subrepresentations. The image of is the eigenspace generated by , so is injective and its image is the subspace of cut out by the same relations (where is the torsor operation on , resp. the automorphism of , indexed by ) that cut out in .
As to part c, since has three surjective characters whose product is , we have with the map given by multiplying the coordinates. Since also injects diagonally into , we easily get a direct sum decomposition, which shows that is injective. As to the image, it is not hard to show that the diagram
commutes, establishing the desired norm characterization of .
The formulas by radicals for the cubic and quartic algebras corresponding to a Kummer element follow easily by chasing through the Galois actions on the appropriate étale algebras. The quartic case is also considered by Knus and Tignol, where a closely related description of is given ([QuarticExercises], Proposition 5.13). ∎
Remark 5.5.
Though it will not be used in the sequel, it is worth noting that Artin-Schreyer theory is amenable to the same treatment.
Theorem 5.6.
Let Let be a finite Galois module with underlying abelian group of exponent .
-
a
There is a natural map
-
b
If , then is injective, is naturally a -torsor, and
-
c
If and , then is injective and
5.1 The Tate pairing and the Hilbert symbol
Assume now that is a local field. Our next step will be to understand the (local) Tate pairing, which is given by a cup product
As we were able to parametrize the cohomology groups in favorable cases, it should not come as a surprise that we can often describe the Tate pairing with similar explicitness.
Recall the definitions of the Artin and Hilbert symbols. If has trivial -action, then , and we have a Tate pairing
Now parametrizes -torsors, while by Kummer theory, . The Tate pairing in this case is none other than the Artin symbol (or norm-residue symbol) which attaches to a cyclic extension , of degree dividing , a mapping whose kernel is the norm group (see Neukirch [NeukirchCoho], Prop. 7.2.13). If, in addition, , then is also isomorphic to , and the Tate pairing is an alternating pairing
classically called the Hilbert symbol (or Hilbert pairing). It is defined in terms of the Artin symbol by
| (16) |
In particular, if and only if is the norm of an element of . This can also be described in terms of the splitting of an appropriate Severi-Brauer variety; for instance, if , we have exactly when the conic
has a -rational point. See also Serre ([SerreLF], §§XIV.1–2). (All identifications between pairings here are up to sign; the signs are not consistent in the literature and are totally irrelevant for this paper.) Pleasantly, for the types of featured in Theorem 5.4, the Tate pairing can be expressed simply in terms of the Hilbert pairing.
We extend the Hilbert pairing to étale algebras in the obvious way: if , then
Note that if is a norm from to , then , but the converse no longer holds. We then have the following:
Theorem 5.7 (a formula for the local Tate pairing).
Let be a local field. For , as in Theorem 5.4, let be the Tate dual of , and let be the corresponding étale algebra, corresponding to the -set of elements of maximal order in , just as corresponds to . The Tate pairing
can be described in terms of the Hilbert pairing in the following cases:
-
(a)
If , then both and embed naturally into , and the Tate pairing is the restriction of the Hilbert pairing on .
-
(b)
If , then we have natural isomorphisms , , and the Tate pairing is the restriction of the Hilbert pairing on .
Proof.
In case b, set . We will do the two cases largely in parallel.
Let denote the set of surjections between two groups . Note that if are Galois modules, then is a -set. Note that is the étale algebra corresponding to the -set
There is an obvious map given by projection to the first factor, which allows us to recover the identification . There is also a map of -sets
which sends a pair (where , ) to the unique surjective satisfying
This allows us to embed into . It is worth noting that when , carries no information and .
Let be the field factors of ; each corresponds to an orbit on . Let Then for , ,
Since (a standard fact), we have
where is the evaluation map. We now apply the following lemma, which slightly generalizes results seen in the literature.
Lemma 5.8.
Let be a subgroup of finite index. Let and be -modules, and let be a map that is -linear (but not necessarily -linear). Denote by the -linear map
Let . Then
Proof.
Since we are concerned with the equality of a pair of -functors, we can apply dimension shifting to assume that . The proof is now straightforward. ∎
Applying with , , and , we get
where and are given by the natural action. Now the outer sum runs over all -orbits of while the inner sum runs over the elements of each orbit, so we simply get
Since the Tate pairing is given by
it remains to check that
as maps from to . In the case , each term is actually equal to , and there are mod terms. In the case , a direct verification on a basis of is not difficult. ∎
6 Rings over a Dedekind domain
Thus far, we have been considering étale algebras over a field . We now suppose that is the fraction field of a Dedekind domain (not of characteristic ), which for us will usually be a number field or a completion thereof, although there is no need to be so restrictive. Our topic of study will be the subrings of that are lattices of full rank over —the orders, to use the standard but unfortunately overloaded word.
There is always a unique maximal order , the integral closure of in . If is a product of field factors, we have .
6.1 Indices of lattices
There is one piece of notation that we explain here to avoid confusion. If is an -dimensional vector space over and are two full-rank lattices, we denote by the index the unique fractional ideal such that
as -submodules of the top exterior power . Alternatively, if , then the classification theorem for finitely generated modules over lets us write
and the index equals
The index satisfies the following basic properties:
-
•
;
-
•
If is a vector space over both and a finite extension , and and are two -sublattices, then ;
-
•
If is a -algebra and , then .
Despite the apparent abstractness of its definition, the index is not hard to compute in particular cases: localizing at a prime ideal, we can assume is a PID, and then it is the determinant of the matrix expressing any basis of in terms of a basis of .
If is a -algebra, is an order, and is a fractional ideal, the index is called the norm of and will be denoted by or, when the context is clear, by . Note the following basic properties:
-
•
If is principal, then .
-
•
If and are two -ideals and is invertible, then . This is easily derived from the theorem that an invertible ideal is locally principal (Lemma LABEL:lem:inv=pri). It is false for two arbitrary -ideals.
-
•
If or , then for any and , the norm of an integral ideal is the ideal generated by the absolute norm
6.2 Discriminants
As is standard, we define the discriminant ideal of an order in an étale algebra to be the ideal generated by the trace pairing
| (17) | ||||
The trace pairing is nondegenerate, that is, (this is one equivalent definition of étale). The primes dividing are those at which is ramified and/or is nonmaximal. This notion is standard and widely used. However, it does not quite extend the (also standard) notion of the discriminant of a -algebra over , which has a distinction between positive and negative discriminants. The Ohno-Nakagawa theorem involves this distinction prominently; Dioses [Dioses] and Cohen–Rubinstein-Salzedo–Thorne [CohON] each frame their extensions of O-N in terms of an ad-hoc notion of discriminant that incorporates the splitting data of an order at the infinite primes. Here we explain the variant that we will use.
Since the trace pairing is alternating in the ’s and also in the ’s, it can be viewed as a bilinear form on the rank- lattice . Identifying with a (fractional) ideal of (whose class is often called the Steinitz class of ), we can write
for some nonzero . Had we rescaled the identification by , would be multiplied by . We call the pair , up to the equivalence , the discriminant of and denote it by .
There is another perspective on the discriminant . Let be the -torsor corresponding to , which comes with embeddings freely permuted by the -action (not to be confused with the coordinates of ). Noting that, for any ,
we can factor the trace pairing matrix:
Define
so that
Now look more carefully at the map . First, is alternating under permutations of the ’s, so it defines a linear map
Moreover, is alternating under postcomposition by the torsor action of on , which permutes the freely. Thus the image of lies in the -torsor , which we call the discriminant torsor of , and even more specifically in the -eigenspace of the nontrivial element of . By (the simplest case of) Kummer theory, we may write . If is any -basis of , so that corresponds to some nonzero element , then
Thus . We summarize this result in a proposition.
Proposition 6.1.
If is an étale algebra over of discriminant , then is the discriminant torsor of ; that is, the diagram of Galois structure maps
commutes.
There is notable integral structure on as well.
Lemma 6.2 (Stickelberger’s theorem over Dedekind domains).
If is the discriminant of an order , then mod for some .
Remark 6.3.
When , Lemma 6.2 states that the discriminant of an order is congruent to or mod : a nontrivial and classical theorem due to Stickelberger. Our proof is a generalization of the most familiar one for Stickelberger’s theorem, due to Schur [Schur1929].
Proof.
Since , the conclusion can be checked locally at each prime dividing in . We can thus assume that is a DVR and in particular that . Now there is a simple tensor that corresponds to the element . By definition,
| (18) |
where
lies in by symmetry and is its conjugate. By construction, is integral over , that is to say, and lie in . Now
is the sum of a square and a multiple of in . ∎
Remark 6.4.
One can write (18) in the suggestive form
where are the two automorphisms of . This equates discriminants of orders in with those of orders in . Equalities of determinants of this sort reappear in Bhargava’s parametrizations of quartic and quintic rings and appear to be a common feature of many types of resolvent fields.
We can now state the notion of discriminant as we would like to use it.
Definition 6.5.
A discriminant over is an equivalence class of pairs , with and mod for some , up to the equivalence relation
If is an étale order, the discriminant is defined as follows: Pick any representation of the Steinitz class as an ideal class; then is the unique pair such that
Note the following points.
-
•
The discriminant recovers the discriminant ideal via .
-
•
If has degree , the discriminant also contains the splitting information of at the infinite primes. Namely, for each real place of , if then , while if then .
-
•
By a usual abuse of language, if is an étale algebra over a number field , its discriminant is the discriminant of the ring of integers over .
-
•
The discriminants over form a cancellative semigroup under the multiplication law
-
•
If is a PID, then we can take , and then the discriminants are simply nonzero elements congruent to a square mod , up to multiplication by squares of units.
-
•
We will often denote a discriminant by a single letter, such as . When elements or ideals of appear in discriminants, they are to be understood as follows:
(19) (20) The seemingly counterintuitive convention (20) is motivated by the fact that, if is principal, then is the same discriminant as .
With these remarks in place, the reader should not have difficulty reading and proving the following relation:
Proposition 6.6.
If are two orders in an étale algebra , then
6.3 Quadratic rings
We will spend a lot of time investigating the number of rings over of given degree and discriminant . For quadratic rings, the problem has a complete answer:
Proposition 6.7 (the parametrization of quadratic rings).
Let be a Dedekind domain of characteristic not . For every discriminant , there is a unique quadratic étale order having discriminant .
Proof.
Note first that the theorem is true when is a field: by Kummer theory, quadratic étale algebras over are parametrized by , as are discriminants; and it is a simple matter to check that . We proceed to the general case.
For existence, let be given. By definition, is congruent to a square mod , . Consider the lattice
To prove that is an order in , it is enough to verify that for any , and this follows from the computation
and the conditions .
Now suppose that and are two orders with the same discriminant . Their enclosing -algebras , have the same discriminant over , and hence we can identify . Now project each along is an -lattice in , which is a one-dimensional -vector space: indeed, we naturally have , and upon computation, we find that . Consequently . Now, for each , the fiber is of the form for some . The element is integral over and lies in , hence in . Thus . ∎
If is any order in an étale algebra (), the quadratic order having the same discriminant as is called the quadratic resolvent ring of . It embeds into the discriminant torsor , in two conjugate ways. Indeed, it is not hard to show that is generated by the elements
appearing in the proof of Lemma 6.2.
Remark 6.8.
The notion of a quadratic resolvent ring extends to characteristic , being always an order in the quadratic resolvent algebra constructed in Example 4.8. We omit the details.
6.4 Cubic rings
Cubic and quartic rings have parametrizations, known as higher composition laws, linking them to certain forms over and also to ideals in resolvent rings. The study of higher composition laws was inaugurated by Bhargava in his celebrated series of papers ([B1, B2, B3, B4]), although the gist of the parametrization of cubic rings goes back to work of F.W. Levi [Levi]. Later work by Deligne and by Wood [WQuartic, W2xnxn] has extended much of Bhargava’s work from to an arbitrary base scheme. In a previous paper [ORings], the author explained how a representative sample of these higher composition laws extend to the case when the base ring is a Dedekind domain. In the present work, we will need a few more; fortunately, there are no added difficulties, and we will briefly run through the statements and the methods of proof.
Theorem 6.9 (the parametrization of cubic rings).
Let be a Dedekind domain with field of fractions , .
-
a
Cubic rings over , up to isomorphism, are in bijection with cubic maps
between a two-dimensional -lattice and its own Steinitz class, up to isomorphism, in the obvious sense of a commutative square
The bijection sends a ring to the index form given by
-
b
If is nondegenerate, that is, the corresponding cubic -algebra is étale, then the map is the restriction, under the Minkowski embedding, of the index form of , which is
(21) -
c
Conversely, let be a cubic étale algebra over . If is a lattice such that sends into , then there is a unique cubic ring such that, under the natural identifications, .
Proof.
-
a
The proof is quite elementary, involving merely solving for the coefficients of the unknown multiplication table of . The case where is a PID is due to Gross ([cubquat], Section 2): the cubic ring having index form
has multiplication table
(22) For the general Dedekind case, see my [ORings], Theorem 7.1. It is also subsumed by Deligne’s work over an arbitrary base scheme; see Wood [WQuartic] and the references therein.
-
b
This follows from the fact that the index form respects base change. The index form of is a Vandermonde determinant that can easily be written in the stated form.
-
c
We have an integral cubic map , which is the index form of a unique cubic ring over . But over , is isomorphic to the index form of . Since (as a cubic ring over ) is determined by its index form, we obtain an identification for which , the projection of onto , coincides with . The uniqueness of is obvious, as must lie in the integral closure of in .
∎
In this paper we only deal with nondegenerate rings, that is, those of nonzero discriminant, or equivalently, those that lie in an étale -algebra. Consequently, all index forms that we will see are restrictions of (21). When cubic algebras are parametrized Kummer-theoretically, the resolvent map becomes very explicit and simple:
Proposition 6.10 (explicit Kummer theory for cubic algebras).
Let be a quadratic étale algebra over (), and let
be the cubic algebra of resolvent (the Tate dual of ) corresponding to an element of norm in Theorem 5.4b, where
so maps bijectively onto the traceless plane in . Then the index form of is given explicitly by
| (23) | ||||
where we identify
using the fact that is the discriminant resolvent of .
Proof.
Direct calculation, after reducing to the case . ∎
Theorem 6.11 (self-balanced ideals in the cubic case).
Let be a Dedekind domain, , and let be a quadratic étale extension. A self-balanced triple in is a triple consisting of a quadratic order , a fractional ideal of , and a scalar satisfying the conditions
| (24) |
-
a
Fix and with a cube . Then the mapping
(25) defines a bijection between
-
•
self-balanced triples of the form , and
-
•
subrings of the cubic algebra corresponding to the Kummer element , such that is -traced, that is, for every .
-
•
-
b
Under this bijection, we have the discriminant relation
(26)
Proof.
The mapping defines a bijection between lattices and . The difficult part is showing that fits into a self-balanced triple if and only if is the projection of a -traced order . Note that if exists, it is unique, as the requirement pins down .
Rather than establish this equivalence directly, we will show that both conditions are equivalent to the symmetric trilinear form
taking values in .
In the case of self-balanced ideals, this was done over by Bhargava [B1, Theorem 3]. Over a Dedekind domain, it follows from the parametrization of balanced triples of ideals over [ORings, Theorem 5.3], after specializing to the case that all three ideals are identified with one ideal . It also follows from the corresponding results over an arbitrary base in Wood [W2xnxn, Theorem 1.4].
In the case of rings, we compute by Proposition 6.10 that is the trilinear form attached to the index form of . By Theorem 6.9c, the diagonal restriction takes values in if and only if lifts to a ring . We wish to prove that itself takes values in if and only if is -traced. Note that both conditions are local at the primes dividing and , so we may assume that is a DVR. With respect to a basis of and a generator of , the index form of has the form
If this is the diagonal restriction of , then itself can be represented as a -dimensional matrix
which is integral exactly when . Since the trace ideal of is generated by
(by reference to the multiplication table (22)), this is also the condition for to be -traced, establishing the equivalence.
The discriminant relation (26) follows easily from the definition of . ∎
6.5 Quartic rings and their cubic resolvent rings
The basic method for parametrizing quartic orders is by means of cubic resolvent rings, introduced by Bhargava in [B3] and developed by Wood in [WQuartic] and the author in [ORings].
Definition 6.12 ([ORings], Definition 8.1; also a special case of [WQuartic], p. 1069).
Let be a Dedekind domain, and let be a quartic algebra over . A resolvent for (“numerical resolvent” in [ORings]) consists of a rank- -lattice , an -module isomorphism , and a quadratic map such that there is an identity of biquadratic maps
| (27) |
from to .
We collect some basic facts about these resolvents.
Theorem 6.13 (the parametrization of quartic rings).
The notion of resolvent for quartic rings has the following properties.
-
a
If is a rank- -lattice and , satisfy (27), then there is a unique (up to isomorphism) quartic ring equipped with an identification making a resolvent.
-
b
There is a canonical (in particular, base-change-respecting) way to associate to a resolvent a cubic ring and an identification with the following property: For any element and any lift of the element , we have the equality
It satisfies
(Here the discriminants are to be seen as quadratic resolvent rings, as in [ORings]; this implies the corresponding identity of discriminant ideals.) If is nondegenerate, then is unique.
-
c
Any quartic ring has at least one resolvent.
-
d
If is maximal, the resolvent is unique (but need not be maximal).
-
e
The number of resolvents of is the sum of the absolute norms of the divisors of the content of , the smallest ideal such that for some order .
-
f
Let be a resolvent of with associated cubic ring , and let . If the corresponding quartic -algebra is étale, then the cubic -algebra is none other than the cubic resolvent of , as defined in Example 4.7. The maps and are the restrictions, under the Minkowski embedding, of the unique resolvent of , which is with the maps
(28) and
(29) -
g
Conversely, let be a quartic étale algebra over and its cubic resolvent. Let
be the resolvent data of as a (maximal) quartic ring over . Suppose , are lattices such that
-
•
sends into ,
-
•
maps isomorphically onto .
Then there are unique quartic rings , such that, under the natural identifications, , , and is a resolvent with the restrictions of and .
-
•
Proof.
-
a
See [ORings], Theorem 8.3.
-
b
See [ORings], Theorems 8.7 and 8.8.
-
c
See [ORings], Corollary 8.6.
-
d
This is a special case of the following part.
-
e
See [ORings], Corollary 8.5.
-
f
By base-changing to , we see that is a resolvent for . Since the resolvent is unique, it suffices to show that the cubic resolvent from Example 4.7 is a resolvent for also. The maps and defined in the theorem statement are seen, by symmetry, to restrict to maps of the appropriate -modules. The verification of (27) and of the fact that the multiplicative structure on is the right one can be checked at the level of -algebras.
-
g
Letting , in part a, we construct the desired and . By comparison to the situation under base-change to , we see that , naturally inject into , respectively. Uniqueness is obvious, as must lie in the integral closure .
∎
In this paper we only deal with nondegenerate rings, that is, those of nonzero discriminant, or equivalently, those that lie in an étale -algebra. Consequently, all resolvent maps , that we will see are restrictions of (28) and (29). When quartic algebras are parametrized Kummer-theoretically, the resolvent map becomes very explicit and simple:
Proposition 6.14 (explicit Kummer theory for quartic algebras).
Proof.
Remark 6.15.
The datum of a resolvent carries no information, in the following sense. It is unique up to scaling by , and the resolvent data and are isomorphic under multiplication by on and by on . If is a PID, indeed, neither nor carries any information, and the entire data of the resolvent is encapsulated in , a pair of symmetric matrices over (with formal factors of off the diagonal) defined up to the natural action of . This establishes the close kinship with Bhargava’s parametrization of quartic rings in [B3]. However, it is useful to keep around.
6.5.1 Traced resolvents
Just as we found it natural to study not just binary cubic -forms, but also -forms and their analogue for each divisor of the ideal , so too we study not just quartic rings in general but those satisfying a natural condition at the primes dividing .
Definition 6.16.
Let be a Dedekind domain, , and let be an ideal dividing in . A resolvent over is called -traced if, for all and in , the associated bilinear form
whose diagonal restriction is takes values in . If is a PID, this is equivalent to saying that the off-diagonal entries in the matrix representation of , which a priori live in , actually belong to . We say that is -traced if it admits a -traced resolvent.
Here are some facts about traced resolvents:
Proposition 6.17.
Let be a quartic ring over a Dedekind domain .
-
is -traced if and only if
-
for all ;
-
for all .
-
-
If is not an order in the trivial algebra , the number of -traced resolvents of is the sum of the absolute norms of the divisors of its -traced content, which is the smallest ideal such that and is also -traced.
-
If is a -traced resolvent with associated cubic ring , then , that is, for some cubic ring . We call a “reduced resolvent” of the -traced ring . Also, .
Proof.
-
Since both statements are local at the primes dividing , we can assume that is a DVR, and thus that is principal. With respect to bases for and for a resolvent , the structure constants of the ring , defined by
are determined by the entries of the resolvent
via the determinants
and a set of formulas appearing in Bhargava [B3, equation (21)] and over a Dedekind domain by the author [ORings, equation (12)]:
(32) where denotes any permutation of and its sign. (Here the nonappearance of some of the individual on the left-hand side of (32) stems from the ambiguity of translating each by , which does not change the matrix of .)
Assume first that is -traced. Then
(33) We then prove that the conditions ai and aii must hold:
-
The trace
and likewise .
-
The coefficients of satisfy:
and likewise for ; and then also, since the trace . So the desired relation holds when , indeed for any . The same proof works for or . Since the case is trivial and squaring is a -linear operation modulo , we get the result for all .
Conversely, suppose that ai and aii hold. We first establish (33). We have
-
•
-
•
-
•
.
Permuting the indices as needed, this accounts for all the about which (33) makes a nontrivial assertion.
Now we work from the back to the resolvent . We may assume that is nontrivial (the trivial rings, one for each Steinitz class, are plainly -traced with .) Then, in the proof of [ORings], Theorem 8.4, the author established that there are vectors in a two-dimensional vector space over , unique up to , such that
for some fixed generator . (The proof uses the Plücker relations, which are a consequence of the associative law on .) This is none other than , the resolvent module of the quartic algebra , which admits the unique resolvent
(34) The resolvents of were found to be exactly the lattices containing the span of the six , with the correct index
the content ideal of . By inspection of (34) that is -traced if and only if it actually contains the span of the six vectors
Condition (33) is interpreted as saying that the are still integer multiples of . Then the -traced resolvents are the lattices . The needed index
is an integral ideal, so such exists, finishing the proof of a.
-
-
It suffices to prove that is the -traced content of . To see this, note that if has content divisible by , then the structure coefficients of are obtained from those of by dividing by . This means that the and are divided by , and so remain integral (indicating that is also -traced) exactly when .
-
We can again reduce to the case that is a DVR so has an -basis. Recall that the index form of the resolvent is given by
([B3], Proposition 11; [ORings], Theorem 8.7). If and have off-diagonal entries in , it immediately follows that is divisible by , so . Consequently , being quartic in the coefficients of , is divisible by . ∎
Similar to Theorem 6.11, we have the following relation between -traced quartic rings and self-balanced ideals:
Theorem 6.18 (self-balanced ideals in the quartic setting).
Let be a Dedekind domain, , and let be a cubic étale extension. A self-balanced triple in is a triple consisting of a cubic order , a fractional ideal of , and a scalar satisfying the conditions
| (35) |
Fix an order and a scalar with a square . Then the mapping
| (36) |
defines a bijection between
-
•
self-balanced triples of the form , and
-
•
subrings of the quartic algebra corresponding to the Kummer element , such that is -traced with reduced resolvent .
Proof.
The proof is very similar to that of 6.11, so we simply summarize the main points. The linear isomorphism establishes a bijection between lattices and . We wish to prove that is balanced if and only if is the projection of a -traced order with reduced resolvent .
First note that either of these conditions uniquely specifies
the former by the balancing condition , and the latter by the -condition that have discriminant .
Once again, it is difficult to proceed directly, and we instead prove that both conditions are equivalent to the bilinear map
taking values in . ∎
On the self-balanced ideals side, this follows from the parametrization of balanced pairs of ideals by boxes performed over by Bhargava [B2, Theorem 2] and over a general base by Wood [W2xnxn, Theorem 1.4].
On the quartic rings side, the diagonal restriction of is precisely the resolvent of , by Proposition 6.14. That for each expresses the one condition remaining for to lift (by Theorem 6.13g) to a quartic ring with resolvent . Then, by definition, this resolvent is -traced exactly when itself has image in .
7 Cohomology of cyclic modules over a local field
Let be a Galois module with underlying group over a local field (that is, a wild local field of characteristic ). Denote by and , respectively, the -torsors corresponding to the action of on and on
and denote by the torsor operation corresponding to . By Theorem 5.4, Kummer theory gives an isomorphism
| (37) |
Our objective in this section is to understand the group on the right: that is, to describe a basis of it (a generalization of the well-known Shafarevich basis for ) and understand how the Tate pairing respects it. Much of our work parallels that of Del Corso and Dvornicich [DCD] and Nguyen-Quang-Do [Nguyen].
If , we let be the -extension of degree coming from the affine action of on , while we let be the associated -torsor. Owing to the semidirect product structure of , we get a natural decomposition
Using the division algorithm in , we let and the integers such that
We call the level, and the offset, of the -extension or of the coclass . Although these definitions appear strange, they allow us to state concisely the following theorem, which will be the main theorem of this section.
Theorem 7.1 (levels and offsets).
Let be a Galois module with underlying group over a local field with .
-
a
The level of a coclass determines its offset uniquely in the following way:
-
i
If , then .
-
ii
If , then and
where is the Kummer element corresponding to the resolvent -torsor of .
-
iii
If , then .
-
i
-
b
For all , the level space
consisting of coclasses of level at least is a subgroup of .
-
c
.
-
d
For ,
-
e
is the whole of , and
-
f
For , a neighborhood
is a level space whose index is given by
where
-
g
For , with respect to the Tate pairing between and ,
One corollary is sufficiently important that we state it before starting the proof:
Corollary 7.2.
For , the characteristic function of the level space has Fourier transform given by
| (38) |
where .
7.1 Discriminants of Kummer and affine extensions
The starting point for our investigation of discriminants is as follows:
Theorem 7.3.
Let be a local field with , and let be a minimal representative of a class in . The discriminant ideal of the associated Kummer extension is given by
Proof.
One can find an explicit basis for and compute the discriminant. For details, see Del Corso and Dvornicich [DCD, Lemmas 5, 6, and 7]. ∎
In this section, we will prove the following generalization:
Theorem 7.4.
Let be a local field, and let be a -module with underlying group . Let be the -torsor corresponding to the -set , and let be the field factor of . Let be a minimal representative for a class in parametrizing, via Theorem 5.4 a coclass , and let be the corresponding -extension. Then
| (39) |
where is the -torsor corresponding to .
Remark 7.5.
Note that is the extension of an ideal of , since divides .
Proof.
If is not a field, then the image of in lies in a nontransitive subgroup (viewing as embedded in ). It is not hard to show that every nontransitive subgroup of has a fixed point. Moving this fixed point to , we get that , , and , in accord with the second case of the formula.
We may now assume that is a field. Although the extension need not be Galois, we have
a Kummer extension of . Let be a field factor of containing . Then
as extensions of . Note that ; in particular, is prime to . So remains a minimal representative in , and since and must be linearly disjoint, is a field unless . So
We must now relate to . If , then is unramified, so and is unramified as well. In particular, is Galois, so is trivial, is totally split, and the formula again holds.
We are left with the case that . Here , and hence , are totally ramified. We relate their discriminants by the following trick, which also appears in Del Corso and Dvornicich [DCD]. An -basis for is given by
| (40) |
The same elements form an -basis for an order , but their -valuations are , where . Divide each basis element by as many times as possible so that it remains integral. We get a new system of elements
| (41) |
Since is coprime to , these elements have -valuations in some order and thus form an -basis for . We have
and hence
Remark 7.6.
Along the lines of the preceding argument, we can prove the following more general result on discriminants in extensions of coprime degree:
Proposition 7.7.
Let and be two extensions of a local field with . Then
7.2 The Shafarevich basis
We start with the following exposition of the Shafarevich basis theorem. Although this theorem has appeared many times in the literature (see Del Corso and Dvornicich, [DCD], Proposition 6), we include a proof here by a method that will establish some important corollaries for us.
Filter by the subgroups
and let be the projection of onto . Note that for , as the Taylor series for about converges for mod . So
| (42) |
as -vector spaces, and we can produce a basis for by lifting a basis for each of the composition factors on the right-hand side.
Proposition 7.8 (the Shafarevich basis theorem).
Let be a local field, and let . The structure of is as follows:
-
•
If
then has a basis of units of the form , where ranges over an -basis of . We call these generic units.
-
•
if and , then has dimension and is generated by
for any with . We call such a generator an intimate unit, and we let
the distance of an intimate unit to .
-
•
For all other we have .
Proof.
Note that because has order prime to . To compute , where , we must see how many of the congruence classes mod (where ) contain a th power.
Consider a general th power , . Write , . By the binomial theorem,
so
We now perform the needed analysis in each case:
-
•
If and , then can never attain the value , so the map is an isomorphism, and we must include an entire basis of elements in our basis for .
-
•
If and , then the th powers
cover all the desired congruence classes, as the map on is surjective; so .
-
•
If , then the th powers of elements of the form surject onto the congruence classes, repeating what we knew from the Taylor series.
-
•
Finally, if , then we can only use powers where . We have
where
So we must analyze the (clearly linear) map given by .
If is not a st power in (or in , which amounts to the same thing by Hensel’s lemma), then is injective and hence surjective, so . Also, has no nontrivial th roots of unity, as would yield a nontrivial element of (since ).
If is a st power in , then is an element of . Note that lifts to a nontrivial th root of unity , since mod has (by the Taylor series again) a th root that is mod . Note that has dimension only , since is unique up to . Consequently has dimension exactly .
This does not tell us how to find a generator for . For this, put so
where is the usual Artin-Schreyer map. Since and are Galois conjugates over , we have , so if is an element with nonzero trace to , then is generated by
We draw two corollaries of the above method.
Corollary 7.9.
has dimension
with a basis consisting of and (arbitrary lifts of) the elements in the bases of in Proposition 7.8.
We call this basis the Shafarevich basis for .
Remark 7.10.
The dimension of follows also from the Euler-characteristic computation
Proof.
Clearly is generated by . The result follows from the composition series (42). ∎
Corollary 7.11.
An element belongs to if and only if, in the expansion of in the Shafarevich basis, only the basis elements in appear (to nonzero exponents).
Proof.
Filtering by the , for , we see that a basis for is given by the portion of the Shafarevich basis coming from for . This is just the basis elements that lie in . ∎
The following simple result is one I have not seen in the literature before:
Corollary 7.12.
The cyclotomic extension is isomorphic to and has Kummer element as a -torsor.
Proof.
Let , with uniformizer . In the notation of the intimate unit case of Proposition 7.8, we have , ,
and we can take . Accordingly, is congruent mod to a unique th root of unity . Note that mod . Since acts on by multiplication, it must act on the powers of by . Hence as -torsors. ∎
In the rest of this section we will study how behaves under field extension. We use the following notational conventions:
Elements are classified by their distance, by which we mean the closest distance of a representative from :
Here the absolute value is the local one on . (We could choose a normalization of this absolute value, but we prefer to express in terms of an undetermined and .) Note that , since can always be taken to have nonnegative valuation. Also, it is easy to see that , so defines a norm on .
For ease in stating theorems involving distances, we note that an ideal (or even a fractional ideal) of a local field is uniquely determined by the largest absolute value of its elements, which we denote by . We have
For any real , let denote the closed ball of radius about in :
and likewise for . It is easy to prove that these are subgroups. If is an ideal, then is the projection of ; but this fails for .
Note the following:
Lemma 7.13.
If is an extension of local fields whose degree is prime to , then the canonical map from to is injective and preserves distance: that is, for every ,
Proof.
The injectivity follows from the fact that if , then and is prime to .
It is obvious that , so it suffices to prove the opposite inequality. It’s easy to see that if , then
Let achieve . Then
But is a power of up to th powers, so , completing the proof. ∎
Assume . We first parametrize itself. By (classical) Kummer theory, we have canonical isomorphisms
where the isomorphism is given by Teichmüller lift. (Note that we do not need to pick a generator of to do this.) We have the following:
Lemma 7.14.
If is the Galois module with underlying group corresponding to the Kummer element , then the Tate dual has Kummer element .
Proof.
When is trivial, the result was proved as Corollary 7.12. The lemma then follows by noting that if , are cyclic Galois modules of order with Kummer elements , , respectively, then is also cyclic of order and has Kummer element . ∎
If has field factors (all necessarily isomorphic to one ), then, by Corollary 7.9,
The group is a representation of over the field . Since has the st roots of unity, such a representation splits as a direct sum of -dimensional representations; there are of these, and they are the powers of the standard representation
given by the obvious isomorphism. By Theorem 5.4, is parametrized by the -isotypical component of , which we denote by for brevity.
We can reduce the problem from to in the following way:
Lemma 7.15.
Let be a -torsor, . Let be the field factor of , and let . Then:
-
a
The subgroup fixing (as a set) is , and is a -torsor;
-
b
Projection to defines an isomorphism , where
(43) (44) -
c
More generally, for any with , the orbit consists of field factors whose product is a -torsor. Projection onto and then onto defines isomorphisms
Remark 7.16.
A result with much the same content, but in a slightly different setting, is proved by Del Corso and Dvornicich ([DCD], Proposition 7).
Proof.
The torsor action must permute the field factors transitively; since is cyclic, a generator must cyclically permute them, and the stabilizer of (as a set) is , provinga. Since the action simply transitively permutes the coordinates of , is a -torsor. If we know the -component of an , then all the other components are uniquely determined by the eigenvector condition; it is only necessary for to behave properly under , namely that .
We now filter as above to discover its -component.
Proposition 7.17 (the Shafarevich basis for ).
As above, let be the -torsor corresponding to the Tate dual of a cyclic Galois module of order with Kummer element , and let be the field factor of . Filter by the subgroups as in the previous subsection. Since the -torsor action on preserves the valuation, each is a subrepresentation of . Then:
-
a
The -isotypical component of has dimension if is trivial, otherwise.
-
b
The -isotypical component of has dimension
-
•
if
-
•
if and is trivial;
-
•
otherwise.
-
•
Proof.
Note that is a copy of the trivial representation and that is trivial (as a representation of ) exactly when , that is, is trivial.
Our convention for the Kummer map is such that
By a standard result in Kummer theory, the degree is the least integer such that is a th power, and with
Let and . We claim that and are respectively the ramification and inertia indices of over . By the Euclidean algorithm, we may choose integers and such that
| (45) |
Construct the elements
Note that , so
| (46) |
On the other hand, , and is an th root of the unit
Note that is not an th power for any prime , as otherwise would be a th power, contradicting what we know about . So the residue class of generates a degree- extension of inside ; in particular,
| (47) |
Equality must hold in (46) and (47), so has uniformizer and residue field generator . Note that for ,
We now have what we need to compute the Galois action on . By Proposition 7.8, the space is nonzero only for
(the generic units) and possibly for also (the intimate units).
We begin with the first case. Here has a basis
For ,
Thus the basis element generates a -dimensional -submodule of isomorphic to . Accordingly, we select the generic units satisfying
Since , there are exactly values of satisfying this when
| (48) |
and none otherwise. By (45), is the multiplicative inverse of mod , so we can rewrite (48) as
as desired.
As to the case that (the intimate units), we simply note that, by Proposition 7.4, we have
so
(A direct computation of the torsor action on the intimate units is also possible; it turns out that as -modules.) ∎
By complete reducibility, we can get a basis for from the bases for its composition factors:
Corollary 7.18.
If , then has dimension
with a basis consisting of appropriate lifts of the Shafarevich basis elements picked out by Proposition 7.17.
7.3 Proof of Theorem 7.1
It now remains to recast the above results in terms of levels and offsets and prove the remaining parts of Theorem 7.1.
Let , being a minimal-distance element. We consider the possibilities for the leading factor in with respect to the Shafarevich basis; this determines by Corollary 7.9, and thence and hence the level and offset of .
-
•
If is led by the uniformizer, then is trivial. From Theorem 7.4, we get , so and .
-
•
If is led by a generic unit, then we have where is an integer satisfying
and each of these values is attained by some . Using the one-to-one correspondence of Theorem 7.4, we get that attains exactly the values such that
Thus , , and is determined by via the condition mod
-
•
If is led by an intimate unit or , then was already computed in proving Theorem 7.4. Since is a product of tamely ramified extensions, we have , so .
In the case that is led by a generic unit of level , , there are alternative ways to characterize . We have
from which
Since is in bijection with by Theorem 7.4, we can likewise determine
Item f demands that we invert this to determine how the level changes as ranges in a ball
If , then every in this ball is a th power (by Proposition 7.8, or simply by noting that the Taylor series for th root converges on this ball), so the range of is . If , then the intimate units are certainly included, so the range of is at least ; it also includes all generic units whose levels satisfy
Using the exchange
valid for all real numbers and , we can get this into a form solvable for :
So the range of is , where
| (49) |
as claimed in f.
For b, we note that as decreases from to , the corresponding in (49) hits every value from to , since the argument to the outer ceiling increases by jumps of . So each is a subgroup. For d, we note that , while for
has elements, where is the unique value of for values of having level .
Finally, we have claimed a relation g regarding how level spaces interact with the Tate pairing. In the case of the Hilbert pairing, the result we need is as follows:
Lemma 7.19 (an explicit reciprocity law).
Let be a local field with . If satisfy
then the Hilbert pairing vanishes.
Proof.
This is a consequence of the conductor-discriminant formula (see Neukirch [Neukirch], VII.11.9): For a Galois extension ,
where ranges over the irreducible characters of . Here we apply the formula to . Scale by th powers to be as close to as possible. If or is an intimate unit, the Hilbert pairing clearly vanishes since is unramified and is a unit. So we can assume that is a ramified extension of degree . Then there are -many characters on , all of dimension . One is the trivial character, whose conductor is . The others all have the same conductor , so
By Theorem 7.4, we have
so is generated by any element with
Note that is actually an attainable norm of an element of , namely . By the given inequality, which implies that the Hilbert symbol
vanishes. ∎
If , , and , then it is easy to verify that the hypothesis of Lemma 7.19 holds in each field factor of , in which the Hilbert pairing is being computed. Hence
However, since
equality must hold. ∎
7.4 The tame case
If is a tame local field, that is, , the structure of is well known. We put
and observe that Theorem 7.1c, d, e, g and Corollary 7.2 still hold.
The wild function field case admits a similar treatment, but now the number of levels is infinite. We do not address this case here.
Part III Composed varieties
8 Composed varieties
It has long been noted that orbits of certain algebraic group actions on varieties over a field parametrize rings of low rank over , which can also be identified with the cohomology of small Galois modules over . The aim of this section is to explain all this in a level of generality suitable for our applications.
Definition 8.1.
Let be a field and its separable closure. A composed variety over is a quasi-projective variety over with an action of a quasi-projective algebraic group over such that:
-
a
has a -rational point ;
-
b
the -points of consist of just one orbit ;
-
c
the point stabilizer is a finite abelian subgroup.
The term composed is derived from Gauss composition of binary quadratic forms and the “higher composition laws” of the work of Bhargava and others, from which we derive many of our examples.
Proposition 8.2.
-
a
Once a base orbit is fixed, there is a natural injection
by which the orbits parametrize some subset of the Galois cohomology group .
-
b
The -stabilizer of every is canonically isomorphic to .
Proof.
-
a
Let be given. Since there is only one -orbit, we can find such that . For any , also takes to and so differs from by right-multiplication by an element in . Define a cocycle by
It is routine to verify that
-
•
satisfies the cocycle condition and hence defines an element of ;
-
•
If a different is chosen, then changes by a coboundary;
-
•
If is replaced by for some , the cocycle is unchanged;
-
•
If the basepoint is replaced by for some , the cocycle is unchanged, up to identifying with in the obvious way. (This is why we can fix merely a base orbit instead of a basepoint.)
So we get a map
We claim that is injective. Suppose that map to equivalent cocycles , . Let be the associated transformation that maps to . By right-multiplying by an element of , as above, we can remove any coboundary discrepancy and assume that on the nose. That is, for every ,
which can also be written as
Thus, is Galois stable and hence defined over . It takes to , establishing that these points lie in the same -orbit, as desired.
-
•
-
b
If , then the -stabilizer of is of course . We claim that the obvious map
is an isomorphism of Galois modules. We compute, for ,
establishing the isomorphism. In particular, the Galois-stable points are the same at as at . Note the crucial way that we used that is abelian. By the same token, the identification of stabilizers is independent of and is thus canonical. ∎
The base orbit is distinguished only insofar as it corresponds to the zero element . Changing base orbits changes the parametrization minimally:
Proposition 8.3.
The parametrizations corresponding to two basepoints differ only by translation:
under the isomorphism between the stabilizers established in the previous proposition.
Proof.
Routine calculation. ∎
While is always injective, it need not be surjective, as we will see by examples in the following section.
Definition 8.4.
-
a
A composed variety is full if is surjective, that is, it includes a -orbit for every cohomology class in .
-
b
If is a global field, a composed variety is Hasse if for every , if the localization at each place lies in the image of the local parametrization
then also lies in the image of the global parametrization .
8.1 Examples
In this section, is any field not of one of finitely many bad characteristics for which the exposition does not make sense.
Example 8.5.
The group can act on the variety , the punctured affine line, by
There is a unique -orbit. The point stabilizer is , and the parametrization corresponding to this composed variety (choosing basepoint ) is none other than the Kummer map
That is full follows from Hilbert’s Theorem 90.
Example 8.6.
Let be the variety of binary cubic forms over with fixed discriminant . This has an algebraic action of , which is transitive over (essentially because carries any three points of to any other three), and there is a ready-at-hand basepoint
The point stabilizer is isomorphic to , but twisted by the character of ; that is, as sets with Galois action. Coupled with the appropriate higher composition law (Theorem 6.9), this recovers the parametrization of cubic étale algebras with fixed quadratic resolvent by in Proposition 4.18. To see that it is the same parametrization, note that a that takes to is determined by where it sends the rational root of , so the three ’s are permuted by just like the three roots of . In particular, is full.
Example 8.7.
Continuing with the sequence of known ring parametrizations, we might study the variety of pairs of ternary quadratic forms with fixed discriminant . This has one orbit over under the action of the group ; unfortunately, the point stabilizer is isomorphic to the alternating group , which is not abelian.
So we narrow the group, which widens the ring of invariants and requires us to take a smaller . We let alone act on pairs of ternary quadratic forms, which preserves the resolvent
a binary cubic form. We let be the variety of for which is a fixed separable polynomial. These parametrize quartic étale algebras over whose cubic resolvent is fixed. There is a natural base orbit whose associated has a linear factor. The point stabilizer , with the three non-identity elements permuted by in the same manner as the three roots of . We have reconstructed the parametrization of quartic étale algebras with fixed cubic resolvent by in Proposition 4.18. In particular, is full.
Example 8.8.
Alternatively, we can consider the space of binary quartic forms whose invariants , are fixed. The orbits of this space have been found useful for parametrizing -Selmer elements of the elliptic curve , because the point stabilizer is with the Galois-module structure . This space embeds into the space of the preceding example via a map which we call the Wood embedding after its prominent role in Wood’s work [WoodBQ]:
In general, is not full. For instance, over , if has full -torsion, there are only three kinds of binary quartics over with positive discriminant (positive definite, negative definite, and those with four real roots) which cover three of the four elements in . Two of these three (positive definite, four real roots) form the subgroup of elements whose corresponding -torsor is soluble at : these are the ones we retain when studying . The fourth element of yields étale algebras whose has
a conic with no real points. However, over global fields, it is possible to show that is Hasse, using the Hasse-Minkowski theorem for conics.
Remark 8.9.
Because of the extreme flexibility afforded by general varieties, it is reasonable to suppose that any finite -Galois module appears as the point stabilizer of some full composed variety over . However, we do not pursue this question here.
8.2 Integral models; localization of orbit counts
Let be a number field and its ring of integers. Let be a composed variety, and let be an integral model, that is, a pair of a flat separated scheme and a flat algebraic group over acting on it, equipped with an identification of the generic fiber with . Then , and the -orbits on decompose into -orbits.
Lemma 8.10 (localization of global class numbers).
Let be an integral model for a composed variety . For each place , let
be a function on the local orbits, which we call a local weighting. Suppose that:
-
i
is Hasse.
-
ii
has class number one, that is, the natural localization embedding
is surjective.
-
iii
For each place , there are only finitely many orbits of on . This ensures that the weighted local orbit counter
takes finite values. (Here is a representative of the -orbit corresponding to . If there is no such orbit because is not full, we take .)
-
iv
For almost all , consists of at most one orbit in each -orbit, and identically.
Then the global integral points consist of finitely many -orbits, and the global weighted orbit count can be expressed in terms of the by
| (50) |
Proof.
Grouping the -orbits into -orbits, it suffices to prove that for all ,
| (51) |
If there is no , the left-hand side is zero by definition, and at least one of the is also zero since is Hasse. So we fix an . The right-hand side of (51), which is finite by hypothesis iv since is unramified almost everywhere, can be written as
the sum being over systems of such that is -integral. Since has class number one, each such system glues uniquely to a global orbit , for which is -integral for all , that is, -integral. Thus the right-hand side of (51) is now transformed to
Now each corresponds to a term of the left-hand side of (51) under the map
The fiber of each has size
So we match up one term of the left-hand side, having value
with -many elements on the right-hand side. In view of the outlying factor , this completes the proof. ∎
8.3 Fourier analysis of the local and global Tate pairings
We now introduce the main innovative technique of this thesis: Fourier analysis of local and global Tate duality. In structure we are indebted to Tate’s celebrated thesis [Tate_thesis], in which he
-
1.
constructs a perfect pairing on the additive group of a local field , taking values in the unit circle , and thus furnishing a notion of Fourier transform for -valued functions on ;
-
2.
derives thereby a pairing and Fourier transform on the adele group of a global field ;
-
3.
proves that the discrete subgroup is a self-dual lattice and that the Poisson summation formula
(52) holds for all satisfying reasonable integrability conditions.
In this paper, we work not with the additive group but with a Galois cohomology group . The needed theoretical result is Poitou-Tate duality, a nine-term exact sequence of which the middle three terms are of main interest to us:
This can be interpreted as saying that and (where is the Tate dual) map to dual lattices in the respective adelic cohomology groups
which are mutually dual under the product of the local Tate pairings
Here, for a local field, the local Tate pairing is given by the cup product
It is well known that this pairing is perfect. (The Brauer group is usually described as being but, having no need for a Galois action on it, we identify it with to avoid the need to write an exponential in the Fourier transform.) Now, for any sufficiently nice function (locally constant and compactly supported is more than enough), we have Poisson summation
for some constant which we think of as the covolume of as a lattice in the adelic cohomology. (In fact, by examining the preceding term in the Poitou-Tate sequence, need not inject into , but maps in with finite kernel; but this subtlety can be absorbed into the constant .)
We apply Poisson summation to the local orbit counters defined in the preceding subsection and get a very general reflection theorem.
Definition 8.11.
Let be a local field. Let and be a pair of composed varieties over whose associated point stabilizers , are Tate duals of one another, and let be an integral model of . Two weightings on orbits
are called (mutually) dual with duality constant if their local orbit counters are mutual Fourier transforms:
| (53) |
where the Fourier transform is scaled by
An equation of the form (53) is called a local reflection theorem. If the constant weightings are mutually dual, we say that the two integral models are naturally dual.
Theorem 8.12 (local-to-global reflection engine).
Let be a number field. Let and be a pair of composed varieties over whose associated point stabilizers , are Tate duals of one another. Let be an integral model for each , and let
be a local weighting on each integral model. Suppose that each integral model and local weighting satisfies the hypotheses of Lemma 8.10, and suppose that at each place , the two integral models are dual with some duality constant . Then the weighted global class numbers are in a simple ratio:
Proof.
By Lemma 8.10,
At almost all , each is supported on the unramified cohomology, and must be constant there because otherwise its Fourier transform would not be supported on the unramified cohomology. However, cannot be identically because of the existance of a global basepoint. So for such ,
In particular, the product is a locally constant, compactly supported function on , which is more than enough for Poisson summation to be valid.
Since the pairing between the adelic cohomology groups is made by multiplying the local Tate pairings, a product of local factors has a Fourier transform with a corresponding product expansion: