Algebraic varieties are homeomorphic to varieties defined over number fields

Algebraic Varieties over Small Fields

— We study curves and their Jacobians over F̄p and Q̄, and discuss applications to rational connectivity over these fields. We introduce certain dynamical systems on P, induced by translates by torsion points on elliptic curves, and study fields related to these systems.

Modular abelian varieties over number fields

The main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the zeta function L(B/K; s) is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class at...

Abelian Varieties over Large Algebraic Fields with Infinite Torsion

Let A be a non-zero abelian variety defined over a number field K and let K be a fixed algebraic closure of K. For each element σ of the absolute Galois group Gal(K/K), let K(σ) be the fixed field in K of σ. We show that the torsion subgroup of A(K(σ)) is infinite for all σ ∈ Gal(K/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of W.-D. Geyer and ...

Partial Zeta Functions of Algebraic Varieties over Finite Fields

Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil type conjectures. The first approach, using an inductive fibred variety point of vi...

Torsion of abelian varieties over large algebraic fields