Discrepancy operators and numerical integration on compact groups

In this paper we establish an inequality of Koksma-Hlawka-type for compact groups. We first define a discrepancy for compact groups based on discrepancy operators introduced by W. Fleischer and show the relation to the classicalL ²-discrepancy. Then we prove the inequality for functions in a weightedL ²-space.     

An estimate for symmetric square <Emphasis Type="Italic">L</Emphasis>-functions on weight aspect

We establish a mean square estimate on the weight aspect for symmetric square L-functions at every point on the critical line. Received: 15 February 2002     

Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes at integer distances 2, 3, 4, …  to the origin. The faces are considered up to the action of the group of integer-linear transformations.     

Primes in quadratic progressions on average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.     

SOLUTIONS OF THE EQUATION x^m＋y^m=z^m IN SL2z

This paper solves an open problem of Vaserstein. The main result is: the equation x^m y^m=z^m has a solution in SL2z if and only if m is not divisible by 4 or 6 and m is not congruent to 3 modulo 6.     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.     

The Tate-Voloch conjecture for Drinfeld modules

We study the v-adic distance from the torsion of a Drinfeld module to an affine variety.