Related papers: Explicit CM-theory for level 2-structures on abeli…

Generically, one can attach to a Q-curve C octahedral representations Gal(Qbar/Q) --> GL(2,Fbar_3) coming from the Galois action on the 3-torsion of those abelian varieties of GL_2-type whose building block is C. When C is defined over a…

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic…

An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the…

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is…

Let S be a K3 surface. In part I of this paper, we constructed a representation of the group Aut D(S), of auto-equivalences of the derived category of S. We interpreted this infinite dimensional representation, as the natural action of Aut…

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

We define the isomorphism classes of torus-equivariant rank 2 arithmetically Cohen-Macaulay (aCM) vector bundles on the Veronese surface, up to a twist by the hyperplane class, and count them. Our approach makes use of Klyachko's…

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.

We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic…

Let $F$ be a totally real number field of class number one, and let $K$ be a CM-field with $F$ as its maximal real subfield. For each positive integer $N$, we construct a class group of certain binary quadratic forms over $F$ which is…

We describe and illustrate the local neighbourhoods of vertices and edges in the (2, 2)-isogeny graph of principally polarized abelian surfaces, considering the action of automorphisms. Our diagrams are intended to build intuition for…

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to…

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and…

Famously, Kohel proved that isogeny graphs of ordinary elliptic curves are beautifully structured objects, now called volcanos. We prove graph structural theorems for abelian varieties of any dimension with commutative endomorphism ring and…

We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of…

Let $E$ be an ordinary elliptic curve over a finite field and $g$ be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class…

For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer-Manin obstructions. Given a Galois extension of the ground field one can consider similar…

Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is…

We describe how isogenies between principally polarized abelian surfaces (PPAS) with real multiplication (RM) can be decomposed into elementary isogeny types. This leads to a strategy for enumerating the isogeny class of a given PPAS with…