D ec 2 00 7 Character sums to smooth moduli are small

ec 2 00 7 Character sums to smooth moduli are small

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...

ar X iv : a st ro - p h / 05 12 61 7 v 1 2 7 D ec 2 00 5 3 D Spectroscopy of Herbig - Haro objects

un 2 00 7 Profinite complexes of curves , their automorphisms and anabelian properties of moduli stacks of curves

Let M g,[n] , for 2g − 2 + n > 0, be the Deligne-Mumford (briefly D-M) moduli stack of smooth curves of genus g labeled by n unordered distinct points. The main result of the paper is that a finite, connectedétale cover M λ of M g,[n] , defined over a sub-p-adic field k, is " almost " anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces.

D ec 2 00 4 , INCOMPLETE QUADRATIC EXPONENTIAL SUMS IN SEVERAL VARIABLES

We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1 2n ∑

ar X iv : h ep - t h / 06 10 10 2 v 2 1 2 D ec 2 00 6 Flux Compactification

We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies that there is a “landscape” of string/M theory vacua, perhaps containing many candidates for describing r...