Symbol Length in Brauer Groups of Elliptic Curves

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 $${\rho }_{E,\ell }:\text{Gal}(\overline{K}/K)\to {\text{GL}}_2({\mathbb{F}}_{\ell }),$$ 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 ${}_{\ell }\text{Br}(E)/{}_{\ell }\text{Br}(K)$. 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.

15 pages