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

TL;DR

This paper explores the relationship between special values of twisted L-functions and the Galois module structure of Tate-Shafarevich groups, providing conjectures, numerical evidence, and practical methods for performing 11-descents over specific number fields.

Contribution

It introduces a conjecture linking L-values to Tate-Shafarevich groups, and develops a practical procedure for 11-descents over $C_5$-number fields for elliptic curves with complex multiplication.

Findings

Numerical evidence supports the conjecture relating L-values and Tate-Shafarevich groups.

A practical method for 11-descent over $C_5$-number fields is presented.

Potential to extend the method to higher descents like 31-descent.

Abstract

We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character $\chi$, we give an explicit conjecture relating the ideal factorization of $L(E,\chi,1)$ to the Galois module structure of the Tate-Shafarevich group of $E/K$, where $\chi$ 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.