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Shadow line distributions

Jennifer S. Balakrishnan, Mirela Çiperiani, Barry Mazur, Karl Rubin

TL;DR

The paper introduces shadow lines $L_K$ in $E(\mathbb{Q})\otimes\mathbb{Z}_p$, arising from universal norms in the anticyclotomic $\mathbb{Z}_p$-extension along Heegner fields and conjecturally lying in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ when the twisted curve has analytic rank $1$. It develops a computational framework to compute the slopes $s_K$ of these lines via the anticyclotomic $p$-adic height pairing, and analyzes their distribution as the imaginary quadratic field $K$ varies, distinguishing between non-anomalous and anomalous primes $p$. For non-anomalous $p$ the slopes appear equidistributed modulo $p$ (and modulo $p^2$ up to refinements), while for anomalous $p$ distributions are biased due to the receptacle $H$ and potential rational $p$-isogenies; a distinguished mod $p$ shadow line $\mathcal{S}$ often coincides with the natural line $\mathcal{L}$ in $H$, under suitable hypotheses. The authors further show, both empirically and under certain hypotheses, that the projection of shadow lines onto $H$ largely determines the slope behavior, and provide evidence suggesting that the map $K\mapsto L_K$ is injective, i.e., the shadow line encodes information about the source field. Altogether, the work combines Iwasawa-theoretic constructions with extensive computations to illuminate how Heegner points govern the $p$-adic geometry of Mordell–Weil groups.

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free $\mathbb{Z}_p$-submodule of rank $1$, in $ E(K)\otimes \mathbb{Z}_p$ given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic $\mathbb{Z}_p$-extension of $K$; we call it the {\it shadow line}. When the twist of $E$ by $K$ has analytic rank $1$, the shadow line is conjectured to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ as $K$ varies, framing conjectures based on the computations we have made.

Shadow line distributions

TL;DR

The paper introduces shadow lines in , arising from universal norms in the anticyclotomic -extension along Heegner fields and conjecturally lying in when the twisted curve has analytic rank . It develops a computational framework to compute the slopes of these lines via the anticyclotomic -adic height pairing, and analyzes their distribution as the imaginary quadratic field varies, distinguishing between non-anomalous and anomalous primes . For non-anomalous the slopes appear equidistributed modulo (and modulo up to refinements), while for anomalous distributions are biased due to the receptacle and potential rational -isogenies; a distinguished mod shadow line often coincides with the natural line in , under suitable hypotheses. The authors further show, both empirically and under certain hypotheses, that the projection of shadow lines onto largely determines the slope behavior, and provide evidence suggesting that the map is injective, i.e., the shadow line encodes information about the source field. Altogether, the work combines Iwasawa-theoretic constructions with extensive computations to illuminate how Heegner points govern the -adic geometry of Mordell–Weil groups.

Abstract

Let be an elliptic curve over with Mordell--Weil rank and be an odd prime of good ordinary reduction. For every imaginary quadratic field satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free -submodule of rank , in given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic -extension of ; we call it the {\it shadow line}. When the twist of by has analytic rank , the shadow line is conjectured to lie in ; we verify this computationally in all our examples. We study the distribution of shadow lines in as varies, framing conjectures based on the computations we have made.
Paper Structure (10 sections, 2 theorems, 25 equations)

This paper contains 10 sections, 2 theorems, 25 equations.

Table of Contents

  1. Introduction: the shadow lines
  2. Computations, data, and set-up
  3. Distributions for non-anomalous primes $p$
  4. Non-anomalous data mod $p$
  5. Non-anomalous data mod $p^2$
  6. Distributions for anomalous primes $p$
  7. Anomalous data mod $p$
  8. Anomalous data mod $p^2$
  9. The distinguished shadow line modulo $p$ for anomalous primes $p$
  10. The map from imaginary quadratic fields to shadow lines

Key Result

Lemma 4.8

The map ${\overline\psi}: H \otimes \mathbb{F}_p \to {\widehat{E}} (\mathbb{Z}_p)\otimes \mathbb{F}_p$ is non-trivial if and only if the following hold:

Theorems & Definitions (24)

  • Definition 1.1
  • Conjecture 3.1
  • Example 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Lemma 4.8
  • proof
  • Remark 4.10
  • ...and 14 more