July 27, 2000

Hershy Kisilevsky

Title: Distinguishing L-functions by twists

Abstract:

Let A and B be two abelian varieties (elliptic curves) defined over the rationals, Q. If A and B are isogenous, then rk(A(K))=rk(B(K)) for every finite extension K/Q, -- here A(K) is the Mordell--Weil group of K-rational points on A, and rk is the number of copies of Z in A(K). Zarhin has asked if the converse is true. We examine this and analagous questions for elliptic curves over Q, for class numbers of number fields and for abelian varieties over finite fields.