May 2, 2000

Amnon Besser

Title: Mahler measures, complex and p-adic

Abstract:

The Mahler measure is a quantity associated to a (Laurent) polynomial $P(x_1,\ldots,x_n)=\sum a_I x_1^{i_1}\cdots x_n^{i_n}$ where the ij can be both positive and negative. It is defined by

\begin{displaymath}m(P)=(2\pi i)^{-n}\int_{T^n} \log(\vert P(z_1,\ldots,z_n)\vert) (dz_1/z_1)\cdots (dz_n/z_n), \end{displaymath}

where T is the unit circle |z|=1 in C.

The Mahler measure appears in mathematics in various fields. Its origins are in transcendence theory where it is a measure of the complexity of a polynomial. The famous Lehmer conjecture states that 0 is an isolated point for the collection of Mahler measures of polynomials in one variable with integral coefficients. There is a mysterious connection between Mahler measures and Ergodic theory: The Mahler measure of a polynomial P is the entropy of a dynamical system naturally associated to P.

In the early 80's it was discovered that for some polynomials the Mahler measure is a special value of an L-function. This relation was later ``explained'' by Deninger using the Beilinson conjecture by showing that the Mahler measure is related to the Beilinson regulator.

Recently, Rodriguez-Villegas studied Mahler measures of special families of polynomials and got some interesting formulae involving hypergeometric functions and modular forms.

In the talk I will discuss in more detail the different occurrences of the Mahler measure above. I will also say something about recent work with Deninger defining some p-adic analogues of the Mahler measure.