An analogue of Kida’s formula for elliptic curves with additive reduction

Let $p$ be a prime number and $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Iwasawa [Iwa73] studied growth patterns of $p$-primary parts of class numbers in certain infinite abelian Galois towers of number fields. Let $F$ be a number field. Setting $\mu_{p^{\infty}}\subset\bar{\mathbb{Q}}$ to be the $p$-primary roots of unity, we let $F(\mu_{p^{\infty}})$ denote the Galois extension of $F$ that is generated by $\mu_{p^{\infty}}$. There is a unique $\mathbb{Z}_{p}$-extension $F_{\operatorname{cyc}}/F$ which is contained in $F(\mu_{p^{\infty}})$. This is called the cyclotomic $\mathbb{Z}_{p}$-extension of $F$ and the Galois group $\operatorname{Gal}(F_{\operatorname{cyc}}/F)$ is topologically isomorphic to $\mathbb{Z}_{p}$. For a natural number $n$, we define $F_{n}$ to be the subfield of $F_{\operatorname{cyc}}$ such that $\operatorname{Gal}(F_{n}/F)\simeq\mathbb{Z}/p^{n}\mathbb{Z}$. Thus, one has the tower of Galois extensions

Let $e_{n}$ be the exact power of $p$ that divides the class number of $F_{n}$. Iwasawa proved that there exist invariants $\mu=\mu_{p}(F),\lambda=\lambda_{p}(F)\in\mathbb{Z}_{\geq 0}$ and $\nu=\nu_{p}(F)\in\mathbb{Z}$ such that

for all large enough values of $n$. Moreover, Iwasawa conjectured that the $\mu_{p}(F)=0$ for all number fields $F$. The conjecture has been resolved for abelian extensions $F/\mathbb{Q}$ by Ferrero and Washington [FW79].

Let $K$ be a number field and $L/K$ be a finite Galois extension such that $\operatorname{Gal}(L/K)$ is a $p$-group. Kida [Kid80] showed that there is an explicit relationship between the Iwasawa $\mu$- and $\lambda$-invariants for $K_{\operatorname{cyc}}/K$ and those for $L_{\operatorname{cyc}}/L$. This can be viewed as an analogue of the Riemann-Hurwitz formula for Iwasawa invariants. Iwasawa [Iwa81, Theorem 6] later proved a generalization of this result using Galois cohomology. Let $w$ be a prime of $L_{\operatorname{cyc}}$ and $v$ be the prime of $K_{\operatorname{cyc}}$ such that $w|v$. Set $e(w)$ to denote the ramification index of $w$ over $v$. Let $U(L_{\operatorname{cyc}})$ be the group of units of $L_{\operatorname{cyc}}$.

Mazur [Maz72] formulated the Iwasawa theory of elliptic curves with good ordinary reduction at the primes that lie above $p$. Kato [Kat04] proved that for elliptic curves defined over $\mathbb{Q}$, the $p$-primary Selmer groups considered by Mazur were cofinitely generated and cotorsion over the Iwasawa algebra. Hachimori and Matsuno [HM99] proved a generalization of Kida’s formula for Selmer groups of elliptic curves with good ordinary or multiplicative reduction at the primes that lie above $p$. The Iwasawa theory of elliptic curves with additive reduction at primes above $p$ was initially studied by Delbourgo [Del98], who proved that the natural generalization of Kato’s result should hold, provided additional assumptions are satisfied (see Proposition 4.7). It is natural therefore to extend Kida’s formula to elliptic curves with additive reduction (at primes above $p$). Set $\Lambda$ to denote the Iwasawa algebra. Let $E$ be an elliptic curve over a number field $K$ and $L/K$ be a finite Galois extension with Galois group $G\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(L/K)$. The Iwasawa $\mu$- and $\lambda$-invariants associated to the $p$-primary Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$ are denoted by $\mu_{p}(E/K)$ and $\lambda_{p}(E/K)$ respectively. Stated below is our main result.

Note that the first of the above conditions imply that $E$ has potentially good reduction at the primes above $p$. Leveraging Delbourgo’s results for $K=\mathbb{Q}$, we provide explicit conditions for the assumptions of Theorem B to hold, cf. Proposition 4.7 and Corollary 4.9. We also give an explicit example to illustrate these results, see the example following Assumption 5.1 on p. 20.

We then come to our main application, which is to prove density results for the stability of $\mu$ and $\lambda$-invariants in $\mathbb{Z}/p\mathbb{Z}$-extensions of $\mathbb{Q}$. Furthermore, one is also able to derive surprising results concerning rank stability in these extensions. There is considerable interest in rank stability questions for a fixed elliptic curve in prime cyclic extensions, see for instance [DFK04, DFK07, MR18, KR22, Ray23].

Our results are proven via analytic methods, specifically by an application of Delange’s Tauberian theorem (cf. Theorem 5.2). Let $E_{/\mathbb{Q}}$ be an elliptic curve satisfying certain additional conditions (cf. Assumption 5.1). One of these conditions requires that the $p$-primary Selmer group of $E$ over $\mathbb{Q}_{\operatorname{cyc}}$ is cotorsion over $\Lambda$, and the Iwasawa $\mu$- and $\lambda$-invariants are $0$. For instance, for $p=3$, the elliptic curve $E\mathrel{\mathop{\mathchar 58\relax}}y^{2}+y=x^{3}-3x-5$ is shown to satisfy these conditions. Then, we prove an asymptotic lower bound for the number of $\mathbb{Z}/p\mathbb{Z}$-extensions $L/\mathbb{Q}$ such that the $\mu$- and $\lambda$-invariants over for $L_{\operatorname{cyc}}$ remain $0$.

Given a number field $L$, set $\Delta_{L}$ to denote its discriminant. Let $X>0$, and $\mathcal{S}(X)$ be the set of Galois extensions $L/\mathbb{Q}$ with $\operatorname{Gal}(L/\mathbb{Q})\simeq\mathbb{Z}/p\mathbb{Z}$ and such that $|\Delta_{L}|\leq X$. Take $\mathcal{S}_{E}(X)$ to be subset of $\mathcal{S}(X)$ consisting of the extensions for which the following conditions hold

• •

  $\operatorname{Sel}_{p^{\infty}}(E/L)$ is cofinitely generated and cotorsion over $\Lambda$,

• •

  $\mu_{p}(E/L)=0$ and $\lambda_{p}(E/L)=0$.

We note that for $L\in\mathcal{S}_{E}(X)$, it follows from Proposition 3.3 that $\operatorname{rank}E(L)=0$. Thus, the rank remains stable in such extensions $L/\mathbb{Q}$. We prove asymptotic formulae for $N_{E}(X)\mathrel{\mathop{\mathchar 58\relax}}=\#\mathcal{S}_{E}(X)$ respectively. Given two positive real valued functions $f(X)$ and $g(X)$, we write $f(X)\gg g(X)$ to mean that there is a constant $C>0$ such that $Cf(X)>g(X)$ for all sufficiently large $X$.

On the other hand, if we let $M(X)\mathrel{\mathop{\mathchar 58\relax}}=\#\mathcal{S}(X)$, then, there is a constant $c>0$ such that $M(X)\sim cX^{\frac{1}{p-1}}$. This is a special case of Malle’s conjecture [Mal02, Mal04], and this particular result is due to Mäki [M8̈5]. Later, the count was generalized to arbitrary number field bases by Wright [Wri89]. The power of $\log X$ in Theorem C is negative, however, still is very close to $0$ (especially for large values of $p$). On comparing $N_{E}(X)$ with $M(X)$, our result shows that there is a significantly large number of extensions in which the rank remains stable, compared to the total asymptotic. The difference simply lies in the power of $\log X$. We remark that for $p=3$ and $E\mathrel{\mathop{\mathchar 58\relax}}y^{2}+y=x^{3}-3x-5$, the conditions of Theorem C are satisfied.

Including the introduction, the article consists of five sections. In Section 2, we discuss preliminary notions and set up notation. We begin by discussing the algebraic structure of Selmer groups considered over cyclotomic $\mathbb{Z}_{p}$-extensions. In this section, we also recall the results of Hachimori and Matsuno [HM99]. The Section 3 is devoted to the proof of Theorem B. Following this, we discuss the notion of the Euler characteristic of a $\Lambda$-module in Section 4, and recall results of Delbourgo on elliptic curves over $\mathbb{Q}$ with additive reduction at $p$. These results are used in discussing precise conditions for the conditions of Theorem B are satisfied for $K=\mathbb{Q}$. Finally, in Section 5, Theorem C is proven.

Let $p$ be an odd prime number and $K$ be a fixed number field. Denote by $\Sigma_{p}$, the set of primes of $K$ that lie above $p$. Let $E$ be an elliptic curve that is defined over $K$. Denote by $\Sigma_{\operatorname{bad}}$ the set of primes $v$ of $K$ at which $E$ has bad reduction. Let $\Sigma$ be a finite set of primes of $K$ that contains $\Sigma_{\operatorname{bad}}$ and $\Sigma_{p}$. Throughout, we choose an algebraic closure $\bar{K}/K$ as well as $\bar{K}_{v}/K_{v}$ for any prime $v$ of $K$. At each prime $v$ of $K$, choose an embedding $\iota_{v}\mathrel{\mathop{\mathchar 58\relax}}\bar{K}\hookrightarrow\bar{K}_{v}$. We take $\operatorname{G}_{v}$ to denote the absolute Galois group of $K_{v}$. Let $K_{\Sigma}$ be the maximal algebraic extension of $K$ in which all primes $v\notin\Sigma$ are unramified. Note that $K_{\Sigma}$ is a Galois extension of $K$. We set $\operatorname{G}_{K,\Sigma}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(K_{\Sigma}/K)$. For each prime $v\in\Sigma$, we have a natural map

which is induced by $\iota_{v}$.

For $n\in\mathbb{Z}_{\geq 1}$, take $E[p^{n}]$ to be the $p^{n}$-torsion subgroup of $E(\bar{K})$. Set $E[p^{\infty}]$ to be the $p$-primary part of $E(\bar{K})$. The action of $\operatorname{Gal}(\bar{K}/K)$ descends to an action of $\operatorname{G}_{K,\Sigma}$ on $E[p^{\infty}]$ since $\Sigma$ contains the primes above $p$ and the primes at which $E$ has bad reduction. For $i\geq 0$ and any algebraic extension $\mathcal{K}/K$ that is contained in $K_{\Sigma}$, we set

Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers. There is a unique extension $K_{\operatorname{cyc}}/K$ contained in $K(\mu_{p^{\infty}})$ such that $\operatorname{Gal}(K_{\operatorname{cyc}}/K)$ is topologically isomorphic to $\mathbb{Z}_{p}$. This extension is called the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. The only primes of $K$ that ramify in $K_{\operatorname{cyc}}$ are those that lie above $p$, and hence $K_{\operatorname{cyc}}$ is contained in $K_{\Sigma}$. We set $\Gamma$ to denote the Galois group $\operatorname{Gal}(K_{\operatorname{cyc}}/K)$ and choose a topological generator $\gamma$ of $\Gamma$. Given any integer $n\geq 0$, let $K_{n}/K$ be the extension contained in $K_{\operatorname{cyc}}$ such that $[K_{n}\mathrel{\mathop{\mathchar 58\relax}}K]=p^{n}$. This extension is called the $n$-th layer, and one has the following tower of extensions

Identify the Galois group $\operatorname{Gal}(K_{\operatorname{cyc}}/K_{n})$ with $\Gamma^{p^{n}}$ and set $\Gamma_{n}\mathrel{\mathop{\mathchar 58\relax}}=\Gamma/\Gamma^{p^{n}}$. In particular, one may identify $\Gamma_{n}$ with $\operatorname{Gal}(K_{n}/K)$. In the course of this section, we shall introduce certain Selmer groups considered over $K_{\operatorname{cyc}}$. These will be considered as modules over a certain completed group algebra called the Iwasawa algebra $\Lambda$, which has nice properties. We take $\Lambda$ to denote the inverse limit

with respect to natural quotient maps

for $m\geq n\geq 0$. Letting $T$ denote $(\gamma-1)$ we identify $\Lambda$ with the formal power series ring $\mathbb{Z}_{p}\llbracket T\rrbracket$.

Let $\mathcal{K}$ be an algebraic extension of $K$ which is contained in $K_{\Sigma}$. We introduce the $p$-primary Selmer group of $E$ over $\mathcal{K}$. Let $v$ be a prime of $\mathcal{K}$, and consider the Kummer exact sequence of Galois modules

Associated to (2.1), we have the exact sequence in cohomology

Taking the direct limit as $n\rightarrow\infty$, one obtains the following exact sequence

Of particular interest is the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$, which may in fact be identified with the direct limit $\varinjlim_{n}\operatorname{Sel}_{p^{\infty}}(E/K_{n})$. As is well known, $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$ has a canonical $\Lambda$-module structure. Let $\Sigma_{\operatorname{cyc}}$ be the set of primes of $K_{\operatorname{cyc}}$ that lie above $\Sigma$. Note that all primes of $K$ are finitely decomposed in $K_{\operatorname{cyc}}$, hence $\Sigma_{\operatorname{cyc}}$ is finite. Thus, the Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$ is the kernel of the natural restriction map

In this subsection, we introduce the algebraic Iwasawa invariants associated to the $p$-primary Selmer group $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$. A polynomial $f(T)$ with coefficients in $\mathbb{Z}_{p}$ is said to be distinguished if it is monic and all non-leading coefficients are divisible by $p$. Let $M$ and $M^{\prime}$ be finitely generated and torsion $\Lambda$-modules. A $\Lambda$-module homomorphism $\phi\mathrel{\mathop{\mathchar 58\relax}}M\rightarrow M^{\prime}$ is said to be a pseudo-isomorphism if the kernel and cokernel of $\phi$ are finite. It follows from the structure theory of $\Lambda$-modules (cf. [Was97, Chapter 13]) that if $M$ is a finitely generated and torsion $\Lambda$-module, then it is pseudo-isomorphic to a module $M^{\prime}$ of the following form

In (2.2), $s,t$ are non-negative integers, $m_{i},n_{j}$ are positive integers, and $f_{j}(T)$ are irreducible distinguished polynomials. It is understood in the above notation that if $s$ (resp. $t$) is $0$, then the direct sum is empty.

Let $\mathfrak{X}(E/K_{\operatorname{cyc}})$ denote the Pontryagin dual of $\operatorname{Sel}_{p^{\infty}}(E/K_{\operatorname{cyc}})$. Throughout the article, we make the following assumption.

When $E$ has good ordinary reduction at all primes above $p$, it is conjectured by Mazur that Assumption 2.4 holds. This conjecture was settled by Kato and Rubin in the case when $E$ is defined over $\mathbb{Q}$ and $K/\mathbb{Q}$ is an abelian extension. When $E$ has multiplicative reduction at the primes above $p$, it is still conjectured that Assumption 2.4 holds [HM99, Introduction]. On the other hand, for $K=\mathbb{Q}$ and $E$ an elliptic curve with bad additive reduction at $p$, the conjecture was proven by Delbourgo [Del98] under additional hypotheses.

In this subsection, we recall the results of Hachimori and Matsuno [HM99], who prove an analogue of Kida’s formula for elliptic curves $E_{/K}$ with semistable reduction at the primes of $\Sigma_{p}$. Let $L/K$ be a Galois extension of number fields with Galois group $G=\operatorname{Gal}(L/K)$. We assume that $\#G$ is a power of $p$. Let $\Sigma_{\operatorname{add}}$ be the set of primes of $K$ at which $E$ has additive reduction. For a prime $w$ of $L_{\operatorname{cyc}}$, denote by $e(w)=e_{L_{\operatorname{cyc}}/K_{\operatorname{cyc}}}(w)$ the ramification index of $w$ over $K_{\operatorname{cyc}}$. We recall their main result in the case when $E$ has good ordinary reduction at the primes in $\Sigma_{p}$.

Hachimori and Matsuno prove a similar result in the case when $E$ has split multiplicative reduction at all primes in $\Sigma_{p}$, cf. [HM99, section 8].