We describe the moduli space of equivalence classes 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 triples of certain coverings of elliptic curves together with certain 3-torsion Brauer classes on elliptic curves and rational points of these elliptic curves.
Let $H$ be a Hopf algebra and $\sigma$ be a 2-cocycle on $H$. Using the tensor equivalence between the comodule categories of $H$ and of its 2-cocycle twist $H^\sigma$, we study twists of superpotentials, comodule algebras, and their associated universal quantum groups in the sense of Manin. In particular, we show the invariance of Artin--Schelter regularity of comodule algebras that are connected graded under a 2-cocycle twist of an infinite dimensional Hopf algebra. As a consequence, we show that Koszul AS-regular algebras of the same dimension and same Hilbert series are always 2-cocycle twists of each other, when viewed as comodule algebras over Manin's universal quantum groups.
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.
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).
Symbol Length in Brauer Groups of Elliptic Curves. Proceedings of the American Mathematical Society. Forthcoming.
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
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.
Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation. Transformation Groups. 2022.
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.