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