UBC Mathematics Department

Colloquium Abstract: Dr. Fernando Rodriguez Villegas* Princeton University

Modular Mahler Measures

The Mahler measure m(P) of a polynomial P(x_1, \cdots, x_n) is the integral of \log |P| over the torus |x_1|= \cdots |x_n|=1. It appears in many different contexts: as the entropy of a certain dynamical systems; in Arakelov theory and trascendental number theory as a choice of height of varieties at the infinite primes; and in algebraic geometry, as certain type of period integral conjecturally related to special values of L-functions. In this talk we will consider this last aspect, which has its origin in recent work of Deninger and Boyd. It implies very concrete relations between Mahler measures and special values of L-functions that have been checked numerically in hundreds of cases by Boyd. We will show how for certain families of polynomials m(P) can be given in terms of modular forms and discuss the consequences we may draw from this fact.

*Please note that Dr. Villegas is a candidate for a position in the department. Regular faculty are urged to attend this lecture.

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