These notes are an attempt to convey some of the ideas, if not the substance or the details, of the proof of the Weil conjectures by P. Deligne [De1], as far as I understand them, which is to say somewhat superficially – but after all J.-P. Serre (see [Ser]) himself acknowledged that he didn’t check everything. . . What makes this possible is that this proof still contains some crucial steps wh...

For each natural number N , let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet Lfunctions, that the formula holds “on average” in a certain sense. From this they deduced (under ERH) that all but O (x1/2+ ) of the even natural num...

This paper introduces a general method for relating characteristic classes to singularities of a bundle map. The method is based on the notion of geometric atomicity. This is a property of bundle maps α : E → F which universally guarantees the existence of certain limits arising in the theory of singular connections. Under this hypothesis, each characteristic form 8 of E or F satisfies an equat...

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-proje...

We extend a result of Spearman which provides a sufficient condition for elliptic curves of the form y2 = x3 − px, with p a prime, to have Mordell-Weil rank 2. As in Spearman’s work, the condition given here involves the existence of integer points on these curves.

1 Trigonometric Sums 1 1.1 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Kloosterman Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Geometrization 2 2.1 A Lemma on Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Artin-Shreier Sheaves . . . . . . . . . . . . . . . . . ...

Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmüller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones in semi-simple Lie group actions on noncompact symmetric spaces.