Matilde N. Lalín

Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over F q as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random variables taking the value 0 with probability 2/(q + 2) and 1, e2p i/3, e4p i/3 each with probability q/(3(q + 2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.

Higher Mahler measures and zeta functions

We consider a generalization of the Mahler measure of a multi- variable polynomialP as the integral of log k jPj in the unit torus, as opposed to the classical denition with the integral of log jPj. A zeta Mahler mea- sure, involving the integral ofjPj s , is also considered. Specic examples are

Some examples of Mahler measures as multiple polylogarithms

Abstract The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial.

Fluctuations in the number of points on smooth plane curves over finite fields

Abstract In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field F q as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8] .

Mahler measure of some n-variable polynomial families

The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of Lalin (J. Number Theory 103 (2003) 85–108). The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.

ON A CONJECTURE BY BOYD

The aim of this note is to prove the Mahler measure identity m(x + x-1 + y + y-1 + 5) = 6m(x + x-1 + y + y-1 + 1) which was conjectured by Boyd. The proof is achieved by proving relationships between regulators of both curves.

Mahler measure under variations of the base group

Abstract We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lück-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.

Mahler measures and computations with regulators

In this work we apply the techniques that were developed in [M.N. Lalin, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Y. Gon, H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (5) (2004) 425–442].

An algebraic integration for Mahler measure

There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger, and later Rodriguez-Villegas, and Maillot. While Rodriguez-Villegas made this relationship explicit for the two variable case, it is our goal to understand the three variable case and shed some light on the examples with more variables.

On the Mahler measure of resultants in small dimensions

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to special values of L-functions and polylogarithms.