**UBC Mathematics Department**

*http://www.math.ubc.ca*

## 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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