Symbol Length in Brauer Groups of Elliptic Curves

TL;DR

This paper improves bounds on the symbol length in the Brauer group of elliptic curves over fields, providing explicit algorithms for computation and applying to CM elliptic curves with a focus on Galois representations.

Contribution

It refines existing bounds on symbol length in the Brauer group of elliptic curves and offers an explicit computational algorithm, especially for CM elliptic curves.

Findings

Bound on symbol length improved to [L:K]-1 when   does not divide [L:K]

Further reduction of bounds when   contains an element of order d > 1

Abstract

Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not $2,3,$ or $\ell$ containing a primitive $\ell$-th root of unity. For an elliptic curve $E$ over $K$, we consider the standard Galois representation $$\rho_{E,\ell}: \text{Gal}(\overline{K}/K) \rightarrow \text{GL}_2(\mathbb{F}_{\ell}),$$ and denote the fixed field of its kernel by $L$. Recently, the last author gave an algorithm to compute elements in the Brauer group explicitly, deducing an upper bound of $2(\ell+1)(\ell-1)$ on the symbol length in $\mathbin{_{\ell}\text{Br}(E)} / \mathbin{_{\ell}\text{Br}(K)}$. More precisely, the symbol length is bounded above by $2[L:K]$. We improve this bound to $[L:K]-1$ if $\ell \nmid [L:K]$. Under the additional assumption that $\text{Gal}(L/K)$ contains an element of order $d > 1$, we further reduce it to $(1-\frac{1}{d})[L:K]$. In particular, these bounds hold for all CM elliptic curves, in which case we deduce a general upper bound of $\ell + 1$. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.