Symbol length in Brauer groups of elliptic curves

Abstract

Let ℓ \ell be an odd prime, and let K K be a field of characteristic not 2 2 , 3 3 , or ℓ \ell containing a primitive ℓ \ell -th root of unity. For an elliptic curve E E over K K , we consider the standard Galois representation \[ ρ E , ℓ : G a l ( K ¯ / K ) → G L 2 ( F ℓ ) , \rho _{E,\ell }: Gal(\overline {K}/K) \rightarrow GL_2(\mathbb {F}_{\ell }), \] and denote the fixed field of its kernel by L 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 ) 2(\ell +1)(\ell -1) on the symbol length in ℓ B r ( E ) / ℓ B r ( K ) _{\ell }Br(E) / _{\ell }Br(K) . More precisely, the symbol length is bounded above by 2 [ L : K ] 2[L:K] . We improve this bound to [ L : K ] − 1 [L:K]-1 if ℓ ∤ [ L : K ] \ell \nmid [L:K] . Under the additional assumption that G a l ( L / K ) Gal(L/K) contains an element of order d > 1 d > 1 , we further reduce it to ( 1 − 1 d ) [ L : K ] (1-\frac {1}{d})[L:K] . In particular, these bounds hold for all elliptic curves with complex multiplication, in which case we deduce a general upper bound of ℓ + 1 \ell + 1 . We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.