Vojta's Main Conjecture for blowup surfaces

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.

     

Fully maximal and fully minimal abelian varieties

We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.     

Minoration de la hauteur de Néron-Tate sur les surfaces abéliennes

This paper contains results concerning a conjecture made by Lang and Silverman, predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves.     

Abelian surfaces over totally real fields are potentially modular

Publications mathématiques de l'IHÉS - We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence,...     

Beauville Surfaces,Moduli Spaces and Finite Groups

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves.     

Inertia groups and abelian surfaces

This paper classifies the finite groups that occur as inertia groups associated to abelian surfaces. These groups can be viewed as Galois groups for the smallest totally ramified extension over which an abelian surface over a local field acquires semistable reduction. The results extend earlier elliptic curves results of Serre and Kraus.     

Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties, and was then studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.     

Constructing pairing-friendly hyperelliptic curves using Weil restriction

A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become pairing-friendly over a finite extension of Fq. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the Cocks-Pinch and Brezing-Weng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded ρ-value for simple, non-supersingular abelian surfaces.     

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over Fq(t1/d) whose members obtain arbitrarily large rank as d→∞.     

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

On Hitchin's connection

The aim of the paper is to give an explicit expression for Hitchin's connection in the case of stable rank 2 bundles on genus 2 curves. Some general theory (in the algebraic geometric setting) concerning heat operators is developed. In particular the notion of compatibility of a heat operator with respect to a closed subvariety is introduced. This is used to compare the heat operator in the nonabelian rank 2 genus 2 case to the abelian heat operator (on theta functions) for abelian surfaces. This relation allows one to perform the computation; the resulting differential equations are similar to the Knizhnik-Zalmolodshikov equations.

     

Asymmetric and pseudo-symmetric hyperelliptic surfaces

 In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ₂. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric. Received: 15 November 2001     

Derived categories and Kummer varieties

We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.      

On Counting Certain Abelian Varieties Over Finite Fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.     

Polarizations of isotypical components of Jacobians with group action

We outline a method to compute the type of the induced polarization of an abelian subvariety of a canonically polarized Jacobian of a smooth projective curve. The method works for curves of not too big genus admitting a “large” group of automorphisms. Several examples are given.     

Twisted Fourier-Mukai transforms for holomorphic symplectic four-folds

We apply the methods of C a?ld?raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds which are fibred by Lagrangian abelian surfaces. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain Lagrangian fibration and the derived category of twisted sheaves on its ‘mirror’ partner. As a corollary, we extend the original Fourier-Mukai transform to degenerations of abelian surfaces. Another consequence of the general theory is that the holomorphic symplectic four-fold and its mirror are connected by a one-parameter family of deformations through Lagrangian fibrations.