Related papers: Equivalent computational problems for superspecial…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this…
We give a classification of all principally polarized abelian surfaces that admit an $(l,l)$-isogeny to themselves, and show how to compute all the abelian surfaces that occur. We make the classification explicit in the simplest case $l=2$.…
Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness…
We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that…
We study the set of isomorphism classes of polarized superspecial abelian varieties $(A,\lambda)$ of a fixed dimension over $\mathbb{F}_p$ with Frobenius endomorphism $\pi_A=\sqrt{-p}$ and $\ker \lambda =\ker \pi_A$. This set plays an…
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny,…
Let $q$ be an odd power of a prime $p\in \mathbb{N}$, and $\mathrm{PPSP}(\sqrt{q})$ be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over $\mathbb{F}_q$…
Extending the results of [Asian J. Math. 2019], in [Doc. Math. \textbf{21}, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{odd} degree over the…
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…
Belmans, Oberdieck, and Rennemo asked whether natural automorphisms of Hilbert schemes of points on surfaces can be characterized by the fact that they preserve the exceptional divisor of non-reduced subschemes. Sasaki recently published…
We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing…
In this paper we generalize a theorem of Kudla-Rapoport-Yang which gives a formula for the arithmetic degree of the moduli space of CM elliptic curves together with a special endomorphism of a specified degree. Our extension is to the…
We determine the set of geometric endomorphism algebras of geometrically split abelian surfaces defined over $\mathbb{Q}$. In particular we find that this set has cardinality 92. The essential part of the classification consists in…
A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod-$3$ Galois representation associated to a principally polarized abelian…
We study the B-model chiral ring of Calabi-Yau hypersurfaces in Batyrev's mirror construction. The main result is an explicit description of a subring of the chiral ring of semiample regular (transversal to torus orbits) Calabi-Yau…
It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism…
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…
Recently Ibukiyama proves an explicit formula for the number of certain non-principal polarizations on a superspecial abelian surface, extending his earlier work with Katsura for principal polarizations [Compos. Math. 1994]. As a…
We present a complete classification of complex projective surfaces $X$ with nontrivial self-maps (i.e. surjective morphisms $f:X\rightarrow X$ which are not isomorphisms) of any given degree. The starting point of our classification are…