Speaker: David Rohrlich (BU)
Title: Scarcity and abundance of trivial zeros in division towers
Abstract: By a "trivial zero" of an L-function one usually means a zero which is forced on the L-function by its functional equation. For example the negative even integers are trivial zeros of the Riemann zeta function. In this talk we shall be concerned with possible trivial zeros at s = 1 of twisted L-functions of an elliptic curve E, the twists being irreducible self-dual Artin representations of the Galois group of the p-division tower of E for an odd prime p. Let G be the image of the Galois group in GL(2,Zp). We shall focus on the case where the reduction of G modulo p is contained in a Borel subgroup of GL(2,Fp). The question of trivial zeros then leads to a purely group-theoretic problem, namely to count the number of irreducible self-dual representations of G of level pn as n goes to infinity. The application to trivial zeros depends on a happy coincidence: In both the Frobenius-Schur formula and the factorization of an L-function into twisted L-functions, each irreducible representation is weighted by its dimension.