Papers › Symbol Length in Brauer Groups of Elliptic Curves
Symbol Length in Brauer Groups of Elliptic Curves
Mateo Attanasio, Caroline Choi, Andrei Mandelshtam, Charlotte Ure
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Let ℓ be an odd prime, and let K be a field of characteristic not 2,3, or ℓ containing a primitive ℓ-th root of unity. For an elliptic curve E over K, we consider the standard Galois representation ρ_(E,ℓ): Gal(K/K) →GL₂(𝔽_ℓ), 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(ℓ+1)(ℓ-1) on the symbol length in _ℓBr(E) / _ℓBr(K). More precisely, the symbol length is bounded above by 2[L:K]. We improve this bound to [L:K]-1 if ℓ∤[L:K]. Under the additional assumption that Gal(L/K) contains an element of order d > 1, we further reduce it to (1-1/d)[L:K]. In particular, these bounds hold for all CM elliptic curves, in which case we deduce a general upper bound of ℓ+ 1. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.
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