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On the factorization of twisted $L$-values and $11$-descents over $C_5$-number fields

Céline Maistret, Himanshu Shukla

Abstract

We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character $χ$, we give an explicit conjecture relating the ideal factorization of $L(E,χ,1)$ to the Galois module structure of the Tate-Shafarevich group of $E/K$, where $χ$ factors through the Galois group of $K/\mathbb{Q}$. We provide numerical evidence for this conjecture using the methods of visualization and $p$-descent. For the latter, we present a procedure that makes performing an $11$-descent over a $C_5$ number field practical for an elliptic curve $E/\mathbb{Q}$ with complex multiplication. We also expect that our method can be pushed to perform higher descents (e.g. $31$-descent) over a $C_5$ number field given more computational power.

On the factorization of twisted $L$-values and $11$-descents over $C_5$-number fields

Abstract

We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character , we give an explicit conjecture relating the ideal factorization of to the Galois module structure of the Tate-Shafarevich group of , where factors through the Galois group of . We provide numerical evidence for this conjecture using the methods of visualization and -descent. For the latter, we present a procedure that makes performing an -descent over a number field practical for an elliptic curve with complex multiplication. We also expect that our method can be pushed to perform higher descents (e.g. -descent) over a number field given more computational power.
Paper Structure (15 sections, 16 theorems, 52 equations, 2 algorithms)

This paper contains 15 sections, 16 theorems, 52 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Conjectural order of $\Sha(E/K)$
  3. Conjectural Galois module structure of $\Sha(E/K)$
  4. Related work
  5. Structure of the paper
  6. Acknowledgements
  7. Numerical evidence using visualization
  8. Background
  9. Numerical evidence
  10. Numerical evidence using $p$-descent
  11. Background
  12. Elliptic curves with complex multiplication
  13. Computing S-units and S-class groups
  14. Galois module structure of $\Sha(E/K)[p]$
  15. Numerical evidence

Key Result

Theorem 1.4

Let $\chi$ be a primitive Dirichlet character of order $d$ with conductor $\frak{f}_\chi$. Let $E_1/\mathbb{Q}$ and $E_2/\mathbb{Q}$ be two $p$-congruent elliptic curves (i.e., $E_1[p]\simeq E_2[p]$ as Galois modules). For $i\in\{1,2\}$, assume that $E_i$ has good ordinary reduction at $p$, $E_i[p]$

Theorems & Definitions (41)

  • Conjecture 1.1: Birch and Swinnerton-Dyer Conjecture
  • Definition 1.2: Definition 12 in dew
  • Conjecture 1.3
  • Theorem 1.4: Theorem \ref{['th:mainVisu']}
  • Corollary 1.5: Corollary \ref{['co:mainVisu']}
  • Theorem 1.6
  • Theorem 1.7: Theorem 38 in dew
  • Remark 1.8
  • Theorem 2.1: Theorem vis7fisher, Theorem 2.2
  • Theorem 2.2
  • ...and 31 more