Attanasio M, Choi C, Mandelshtam A.

Symbol Length in Brauer Groups of Elliptic Curves

. Submitted.

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(\overline{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 (1−1d)[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.

Huang H, Nguyen VC, Vashaw KB, Veerapen P, Wang X.

A cogroupoid associated to preregular forms

. Submitted.

We construct a cogroupoid given by preregular forms. This yields a Morita-Takeuchi equivalence between Manin's universal quantum groups associated to superpotential algebras. 

Huang H, Nguyen VC, Ure C, Vashaw KB, Veerapen P, Wang X.

Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation

. Submitted.

Let H be a Hopf algebra that is Z-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural GL2-action on the variables. In particular, we show that these GL2 orbits are in bijection with pairs of j-invariant 0 elliptic curves together with 3-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence.

We determine generators and relations in the d-torsion of the Brauer group of an elliptic curve, provided that the d-torsion of the elliptic curve itself is rational over the base field, for any integer d≥2. For q any odd prime, we further give an algorithm to explicitly calculate generators and relations of the q-torsion of the Brauer group of an elliptic curve over any base field of characteristic different from two, three, and q, containing a primitive q-th root of unity. These generators are symbol algebras and the relations arise as their tensor products. We deduce an upper bound on the symbol length of the prime torsion of Br(E)/Br(k).