### Friday November 10, 2006, 1:40 PM, Goldsmith 226

**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,**Z**_{p}). We shall focus on the case where the
reduction of *G* modulo *p* is contained in a Borel subgroup of
GL(2,**F**_{p}). 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
*p*^{n} 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.