L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

L-functions with Large Analytic Rank and Abelian Varieties with Large Algebraic Rank over Function Fields

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

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 ...

On the Rank of Abelian Varieties over Function Fields

Let C be a smooth projective curve defined over a number field k, A/k(C) an abelian variety and (τ, B) the k(C)/k-trace of A. We estimate how the rank of A(k(C))/τB(k) varies when we take a finite Galois k-cover π : C → C defined over k.

Elliptic Curves with Large Rank over Function Fields

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptoti...

Torsion of abelian varieties over large algebraic fields