Symbol Length in Brauer Groups of Elliptic Curves

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.