The groups of points on abelian varieties over finite fields

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f _A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f _A (1 − t).     

Endomorphisms of abelian varieties and points of finite order in characteristic p

An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 737–744, June, 1977.     

Characterization of abelian varieties

We prove that any smooth complex projective variety X with plurigenera P ₁(X)=P ₂(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety. Oblatum 26-V-1999 & 13-VI-2000?Published online: 11 October 2000     

Syzygies of abelian varieties

We prove a conjecture of R. Lazarsfeld on the syzygies (of the homogeneous ideal) of abelian varieties embedded in projective space by multiples of an ample line bundle. Specifically, we prove that if is an ample line on an abelian variety, then satisfies the property as soon as . The proof uses a criterion for the global generation of vector bundles on abelian varieties (generalizing the classical one for line bundles) and a criterion for the surjectivity of multiplication maps of global sections of two vector bundles in terms of the vanishing of the cohomology of certain twists of their Pontrjagin product.

     

Roots of unity and unreasonable differentiation

Archiv der Mathematik - We explore when it is legal to differentiate a polynomial evaluated at a root of unity using modular arithmetic.     

The loci of abelian varieties with points of high multiplicity on the theta divisor

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure. Research of the first author is supported in part by National Science Foundation under the grant DMS-05-55867.     

Formal subgroups of abelian varieties

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers O _k of k. Completing ? along its zero section defines a formal group over O _k . We prove that any formal subgroup of the generic fiber of whose closure in is smooth over an open subset of Spec O _k arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3]. Oblatum 2-V-2000 & 28-XI-2000?Published online: 5 March 2001     

Root numbers of abelian varieties

We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over number fields. Our result applies to arbitrary abelian varieties. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number associated to an abelian variety over a number field and a complex finite-dimensional irreducible representation of with real-valued character is equal to . We also show that our result is consistent with a refined version of the conjecture of Birch and Swinnerton-Dyer.