Counting the number of the twists of certain polarized abelian varieties

We count the number of isomorphism classes of degree d-twists of some polarized abelian varieties of dimension g over finite fields where either g is an odd prime or else $g=2$. This can be seen as a higher dimensional analogue of the counting problem for the case of elliptic curves.     

On the modularity level of modular abelian varieties over number fields

Let f be a weight two newform for Γ₁(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety Af over certain number fields L. The strategy we follow is to compute the restriction of scalars ResL/Q(B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor NL(B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree, we find that NL(B) belongs to Z and , where fL is the conductor of L.     

On a classification of the automorphism groups of polarized abelian surfaces over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.     

Holomorphic symmetric differentials and a birational characterization of abelian varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions

For any $n\ge 2$ we study the group algebra decomposition of an $([\frac{n}{2}]+1)$ -dimensional family of principally polarized abelian varieties of dimension $n$ with an action of the dihedral group of order $2n$ . For any odd prime $p, n=p$ and $n=2p$ we compute the induced polarization on the isotypical components of these varieties and some other distinguished subvarieties. In the case of $n=p$ the family contains a one-dimensional family of Jacobians. We use this to compute a period matrix for Klein’s icosahedral curve of genus 5.     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

On syzygies of abelian varieties

In this paper we prove the following result: Let be a complex torus and a normally generated line bundle on ; then, for every , the line bundle satisfies Property of Green-Lazarsfeld.

     

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.     

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety defined over a number field that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension has no nonzero pseudo-null submodules.

     

GV‐subschemes and their embeddings in principally polarized abelian varieties

We prove that any embedding of a ‐subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens–Griffiths identifying the intermediate Jacobian of a smooth cubic threefold to the Albanese variety of its Fano surface of lines.     

Computing automorphisms of abelian number fields

Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .

     

Families of irreducible principally polarized abelian varieties isomorphic to a product of elliptic curves

For each greater than or equal to two, we give a family of
-dimensional, irreducible principally polarized abelian varieties isomorphic to a product of elliptic curves. This family corresponds to the modular curve .

     

Fixed-point free automorphisms of abelian varieties

An automorphismf of an abelian varietyX is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. It is well known that the elliptic curve withj-invariant 0 is the only elliptic curve admitting a fixed point free automorphism. In this note, this result is extended to abelian varieties of higher dimensions and some connected commutative algebraic groups.Supported by DFG-contract La 318/4 and EC-contract SC1-0398-C(A).