Prym–Tyurin varieties using self-products of groups

M ay 2 00 8 PRYM - TYURIN VARIETIES USING SELF - PRODUCTS OF GROUPS

Given Prym-Tyurin varieties of exponent q with respect to a finite group G, a subgroup H and a set of rational irreducible representations of G satisfying some additional properties, we construct a Prym-Tyurin variety of exponent [G : H ]q in a natural way. We study an example of this result, starting from the dihedral group Dp for any odd prime p. This generalizes the construction of [4] for p...

Selmer Groups and Chow Groups of Self-products of Algebraic Varieties

Let X be a proper flat scheme over the ring of integers of a global field. We show that the Tate conjecture and the finiteness of the Chow group of vertical cycles on self-products of X implies the vanishing of the dual Selmer group of certain twists of tensor powers of representations occurring in the étale cohomology of X.

Two Problems on Varieties of Groups Generated by Wreath Products

Inaccessible Varieties of Groups

A weak but still unsolved version of the first half of Problem 7 in Hanna Neumann's book [5] asks whether the product 1193 of a nontrivial join-irreducible variety VL and an arbitrary variety 93 is necessarily also join-irreducible. One of the results of this paper is a positive answer provided either It is abelian or the infiniterank free groups of IX have no nontrivial abelian verbal subgroup...

Self-products of Elliptic Curves Arising from Reflection Groups

We prove that quotients of complex vector spaces by lattices of maximal rank invariant with respect to irreducible (complex) reflection groups are isogenous to self-products of elliptic curves, and that for reflections groups which are not the complexifications of the Weyl groups of irreducible root systems, they are isomorphic to self-products of mutually isogenous elliptic curves with complex...