Shadow Line Distributions

There were hints of this in the work of Jacobi before, but it was Poincaré in his 1901 paper Sur les propriétés arithmétiques des courbes algébriques [Po01] who pointed out that the set of rational points on an elliptic curve has a natural (abelian) group structure:

Étudions d’abord la distribution des points rationnels sur ces courbes. J’observe que la connaissance de deux points rationnels sur une cubique rationnelle suffit pour en faire connaître un troisième.

Even though Poincaré also suggests in his paper that the group of rational points on an elliptic curve is finitely generated, it took two decades before this was actually proved to be the case¹¹1(Mordell, 1922). Such groups of rational points, a bit later, were called Mordell–Weil groups, since André Weil had generalized Mordell’s Theorem to prove that the group of $K$-rational points of any abelian variety over any number field $K$ is finitely generated., allowing us to focus on the finite fundamental invariant: the rank of the group of rational points.

Almost half a century later, Néron and Tate—by defining the canonical (Néron-Tate) height on any abelian variety over a number field—established a further canonical structure on the Mordell–Weil groups of elliptic curves: the quotient of such Mordell–Weil groups by their torsion subgroups can be viewed canonically, up to orthogonal isometry, as discrete lattices in Euclidean space of dimension equal to their rank. Subsequently, analogous (canonical) $p$-adic height inner products were defined on Mordell–Weil groups for all prime numbers $p$.

All this provides an intricate interlacing structure on what might have initially been regarded to be a simple arithmetic feature of an elliptic curve over $\mathbb{Q}$: its set of rational points. The object of this paper is to consider further arithmetic architecture canonically constructible on this set. Namely, for any elliptic curve $E$ over $\mathbb{Q}$ with Mordell–Weil rank $2$ and for any prime number $p$ we will be defining (canonically) a web consisting of (we conjecture: infinitely many) $\mathbb{Z}_{p}$-lines in $E(\mathbb{Q})\otimes{\mathbb{Z}_{p}}$ coming from the Mordell–Weil behavior of $E$ over a specific set of (we conjecture: correspondingly infinitely many) quadratic imaginary fields.

Let $E/\mathbb{Q}$ be an elliptic curve of analytic rank $2$ and $p$ an odd prime of good ordinary reduction such that the rational $p$-torsion $E(\mathbb{Q})_{p}$ is trivial. Assume that the $p$-primary part of the Shafarevich–Tate group of $E/\mathbb{Q}$ is finite. Then consider an imaginary quadratic field $K$ such that the analytic rank of $E/K$ is $3$ and the Heegner hypothesis holds for $E$, i.e., all primes dividing the conductor of $E/\mathbb{Q}$ split in $K$. We are interested in the subspace of $E(K)\otimes\mathbb{Z}_{p}$ generated by the anticyclotomic universal norms.

To define this space, let $K_{\infty}$ be the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $K_{n}$ denote the subfield of $K_{\infty}$ whose Galois group over $K$ is isomorphic to $\mathbb{Z}/p^{n}\mathbb{Z}$. The module of universal norms with respect to $K_{\infty}/K$ is defined by

where $N_{K_{n}/K}$ is the norm map induced by the map $E(K_{n})\to E(K)$ given by $P\mapsto\underset{\sigma\in\mathrm{Gal}(K_{n}/K)}{\sum}P^{\sigma}$. Let $L_{K}$ denote the $p$-divisible closure of $\mathcal{U}_{K}$ in $E(K)\otimes\mathbb{Z}_{p}$.

By work of Cornut [Co02] (see the Theorem in the Introduction and the discussion after it) and Vatsal [Va03, Theorem 1.4] on the nontriviality of Heegner points we know that $\operatorname{rk}_{\mathbb{Z}_{p}}L_{K}=1$ if the $p$-primary part of the Shafarevich–Tate group of $E/K_{n}$ is finite for every $n\in\mathbf{N}$, see [MR03, Corollary 4.4]. Complex conjugation acts on $E(K)\otimes\mathbb{Z}_{p}$ and it preserves $\mathcal{U}_{K}$. Consequently $L_{K}$ lies in one of the corresponding eigenspaces $E(K)^{+}\otimes\mathbb{Z}_{p}$ and $E(K)^{-}\otimes\mathbb{Z}_{p}$. Observe that $E(K)^{+}\otimes\mathbb{Z}_{p}=E(\mathbb{Q})\otimes\mathbb{Z}_{p}$. Under our assumptions, by work of Skinner–Urban [SU14], Nekovář [Ne01], Gross–Zagier [GZ], and Kolyvagin [Ko90] we know that

Then by the Sign Conjecture [MR03], we expect $L_{K}$ to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$. Our main motivating questions are the following:

In order to identify the shadow line $L_{K}$, we use the fact that $\mathcal{U}_{K}$ lies in the kernel of the anticyclotomic $p$-adic height pairing $\langle\,,\,\rangle:E(K)\otimes\mathbb{Z}_{p}\times E(K)\otimes\mathbb{Z}_{p}\to\mathbb{Z}_{p}$, see [MT83, Proposition 4.5.2]. The use of this pairing forces us to assume that $p$ splits in $K/\mathbb{Q}$ as otherwise the pairing is trivial. Due to the action of complex conjugation, it follows that the restriction of the pairing to either eigenspace $E(K)^{\pm}$ is trivial. Then since $\operatorname{rk}_{\mathbb{Z}_{p}}E(K)^{-}\otimes\mathbb{Z}_{p}=1$ and if $\operatorname{rk}_{\mathbb{Z}_{p}}E(\mathbb{Q})\otimes\mathbb{Z}_{p}=2$, after verifying the non-triviality of the pairing we can deduce that $L_{K}$ equals the kernel of the anticyclotomic $p$-adic height pairing and $L_{K}\subseteq E(\mathbb{Q})\otimes\mathbb{Z}_{p}$, see [BÇLMN] for further details. We are hence able to identify $L_{K}$ by fixing a basis of $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ and using the anticyclotomic $p$-adic height pairing to compute the slope $s_{K}$ of $L_{K}$, see §2.

Now studying the distribution of shadow lines $L_{K}$ can be done by studying the variation of slopes $s_{K}=(x_{K},y_{K})\in\mathbb{P}^{1}(\mathbb{Z}_{p})$, which we can view modulo $p^{n}$ for $n\in\mathbb{N}$. Note that equidistribution of the shadow lines $L_{K}$ corresponds to the statement that the values of $s_{K}$ modulo $p^{n}$ split equally among the $(p+1)p^{n-1}$ options as $n$ grows when we consider all quadratic imaginary fields of conductor up to some reasonable bound satisfying all the conditions that we have set out.

Our computations indicate that the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ depends on the kind of ordinary reduction of the elliptic curve at the prime $p$.

Our computations suggest that in the case when $p$ is a prime of good ordinary non-anomalous reduction, shadow lines are equidistributed in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$, see §3 and Conjecture 3.1. However, this uniformity fails when $p$ is a prime of good ordinary anomalous reduction but then seems to reappear after two rounds of additional restrictions. The first of these restrictions is the consideration of the receptacle $H\subseteq E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ of shadow lines (see (1.3)), as we will now describe.

Let $K^{\prime\prime}/K^{\prime}$ be a cyclic Galois extension of number fields with $[K^{\prime\prime}:K^{\prime}]=p$ and $\wp^{\prime}$ a prime of $K^{\prime}$ dividing $p$ that is ramified in $K^{\prime\prime}/K^{\prime}$. Let $\wp^{\prime\prime}$ be the prime of $K^{\prime\prime}$ lying above $\wp^{\prime}$, and denote by $k_{\wp^{\prime\prime}}$ and $k_{\wp^{\prime}}$ the corresponding residue fields. Note that $k_{\wp^{\prime\prime}}=k_{\wp^{\prime}}$ since $K^{\prime\prime}/K^{\prime}$ is totally ramified at $p$. We have the following diagram:

Hence in the case of an anomalous prime $p$, many shadow lines $L_{K}$ may lie in $H$, see §4. In order to address this obstruction to the potential equidistribution of shadow lines in the case of anomalous primes, we will study the distribution of

We will now describe the next set of restrictions. For any imaginary quadratic field $K$ that produces a shadow line for $E$ (i.e., satisfying the Heegner hypothesis) we get the commutative diagram:

where ${\widehat{E}}$ denotes the formal group of the elliptic curve $E$. Notice that by (1.4) we can derive the map

Since the algebraic rank of $E/\mathbb{Q}$ is $2$, when ${\overline{\psi}}$ is non-trivial²²2Our computations indicate that this condition fails very rarely. (see Lemma 4.8) its kernel is one-dimensional and denoted by

We call $\mathcal{L}$ the natural line modulo $p$.

To our surprise, we find that most shadow lines $L^{\prime}_{K}$ coincide with $\mathcal{L}$, see Conjecture 4.12, and we even find verifiable conditions that appear to guarantee that $L^{\prime}_{K}\equiv\mathcal{L}\pmod{p}$, see Conjecture 5.5. We prove this result under the assumption that the universal norms $\mathcal{U}_{K}$ are not $p$-divisible in the receptacle $H$, see Proposition 5.9. Finally, our data indicates that this is the last obstruction to the equidistribution of shadow lines for elliptic curves $E$ without a rational $p$-isogeny, see Conjecture 4.14.

We conclude our paper by summarizing the evidence that leads us to believe that a shadow line $L_{K}$ uniquely determines its source field $K$, see §6.

Fix $E/\mathbb{Q}$ an elliptic curve of analytic rank $2$ and $p$ an odd prime of good ordinary reduction such that the $p$-torsion of $E(\mathbb{Q})$ is trivial. Consider imaginary quadratic fields $K$ such that

1. (1)

   the Heegner hypothesis holds for $E$ and $K$;

2. (2)

   the analytic rank of the twisted curve $E^{K}/\mathbb{Q}$ is 1 and $p$ splits in $\mathcal{O}_{K}$.

Under these assumptions we expect that the $\mathbb{Z}_{p}$-rank of $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ equals $2$, and that the shadow line $L_{K}$ corresponding to $(E,p,K)$ lies in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$. In order to study the variation of shadow lines $L_{K}$ as $K$ varies, we fix a basis of $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ by choosing two linearly independent points

and then study the variation of the slope $s_{K}$ of $L_{K}$ with respect to this basis.

In order to compute the slope $s_{K}$, for each quadratic field $K$ we choose a non-torsion point

where $E(K)^{-}$ denotes the $-$eigenspace of $E(K)$ under complex conjugation, see [BÇLMN].

Then the slope $s_{K}$ of the shadow line corresponding to $(E,p,K)$ in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ with respect to the basis $\{P_{1},P_{2}\}$ is

where $\langle\cdot,\cdot\rangle$ denotes the anticyclotomic $p$-adic height pairing. Observe that once $K$ is fixed, our choice of $R$ does not affect the slope $s_{K}$.

In this section, $p$ is a prime of non-anomalous good ordinary reduction for $E$, see Definition 1.1. We considered 15 pairs $(E,p)$ and for each pair we computed $s_{K}$, the slope of shadow line $L_{K}$, for around 200-300 quadratic fields $K$, produced using the first 700 Heegner discriminants for the field.

We should mention that there is a small amount of loss in the data: typically we had to skip a small percentage of fields in the range of computation. These corresponded to fields $K=\mathbb{Q}(\sqrt{D})$ for which either

1. (1)

   finding a non-torsion point in $E^{K}(\mathbb{Q})$ was difficult (after carrying out $2$-, $4$-, and $8$-descents in Magma), or

2. (2)

   after the relevant descents were carried out and a non-torsion point was found, the resulting point had coordinates that were too large for our computations in the following sense: One step in the computation of shadow lines is factoring a denominator ideal in the ring of integers $\mathcal{O}_{K}$ of the quadratic field $K$. Since factorization is very difficult, this step had a time limit in place, and some points had coordinates that were so large that this factorization was not completed within the time limit.

We do not expect this loss to bias the resulting distribution in a significant way.

Once we computed slopes, we recorded the value of the slope modulo $p$. For example, consider the distribution of slopes for (997.c1, 3): from computing with 253 fields, we found that

• •

  69 fields produced slope $(0,1)\pmod{3}$,

• •

  64 fields produced slope $(1,1)\pmod{3}$,

• •

  64 fields produced slope $(2,1)\pmod{3}$,

• •

  56 fields produced slope $(1,0)\pmod{3}$.

We summarize this data by recording the ordered list $[69,64,64,56]$. We note that we considered 253 out of 260 eligible quadratic imaginary discriminants $D$ for (997.c1, 3), with $D\geq-4628$ (the other 7 eligible discriminants in this range were skipped for one of the two reasons mentioned above).

In this section, we consider primes $p$ where $E$ has good ordinary anomalous reduction, see Definition 1.1. Repeating the same process as in §3 for anomalous primes $p$ produces visibly different distributions. We started the investigation in the anomalous case considering the position of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ but soon realized that this had to be refined. Here is one example illustrating what we observed:

The modulo $p$ bias is easy to explain. As described in §1, since the anticyclotomic $\mathbb{Z}_{p}$-extension is eventually totally ramified, the module of universal norms reduces to $0$ in $E(\mathbb{F}_{p})\otimes\mathbb{Z}_{p}$. Hence, universal norms lie in the receptacle $H=\ker\big{\{}E(\mathbb{Q})\otimes\mathbb{Z}_{p}\to E(\mathbb{F}_{p})\otimes\mathbb{Z}_{p}\big{\}}\ \subseteq\ E(\mathbb{Q})\otimes\mathbb{Z}_{p}$ which, in the case of anomalous primes $p$, may differ from $E(\mathbb{Q})\otimes\mathbb{Z}_{p}$.

We now record how we compute the receptacle $H$.

We will now display some data about the distribution of the slopes of shadow lines $L^{\prime}_{K}=L_{K}\cap H$ in the receptacle $H$. For clarity, we review our setup and set some notation. Let $Q_{1},Q_{2}$ be generators of $H$ computed as described in Lemma 4.2, and let $R$ be a non-torsion point of $E(K)^{-}$. Then we compute slopes $s^{\prime}_{K}$ of shadow lines $L^{\prime}_{K}$ in $H$:

for each eligible imaginary quadratic field $K$. We record the distribution of slopes of shadow lines modulo $p$ below.

When does the shadow line attached to $(E,K,p)$ differ from $\mathcal{L}$ modulo $p$? It seems there are some subtle issues to consider here (e.g., if $E$ admits a rational $p$-isogeny, then the statistics are not clear). We would like to find conditions that rule out the non-modal values in the distribution.

The following table shows the effect of restricting to filtered data.