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$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Timo Keller, Mulun Yin

Abstract

Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\mathbf{Q}$. As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$.

$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

Abstract

Let be an elliptic curve and be a prime. We prove the -converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation is reducible) when the -Selmer rank is or . The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field where does not have CM similar to those in [CGLS22] and descent to . As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank or .

Paper Structure

This paper contains 35 sections, 28 theorems, 88 equations.

Table of Contents

  1. Statement of the main results
  2. Method of proof and outline of the paper
  3. Relation to previous works
  4. Eisenstein primes
  5. Potentially multiplicative reduction
  6. Higher weight modular forms
  7. Applications.
  8. Conventions.
  9. Acknowledgements
  10. Primes of additive reduction
  11. Reduction types of elliptic curves
  12. Reduction types of modular forms
  13. Primes of additive reduction
  14. Field of semistable reduction
  15. Iwasawa Theory
  16. ...and 20 more sections

Key Result

Theorem B

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ and let $p>2$ be a prime of potentially good ordinary reduction for $E$. Assume $E[p]$ is reducible. Let $r\in\{0,1\}$. Then and so ${\operatorname{rk}}_{\mathbf{Z}}E(\mathbf{Q})=r$ and $\#\Sha(E/\mathbf{Q})<\infty$.

Theorems & Definitions (52)

  • Conjecture A
  • Theorem B
  • Definition 1.2.1: Remark 12.7 in Kato
  • Proposition 1.4.1: Proposition 6.5 in Conrad
  • Lemma 1.4.2
  • proof
  • Example 2.0.2
  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3
  • ...and 42 more