Arithmetic duality theorems for 1-motives over function fields

Arithmetic over Function Fields

These notes accompany lectures presented at the Clay Mathematics Institute 2006 Summer School on Arithmetic Geometry. The lectures summarize some recent progress on existence of rational points of projective varieties defined over a function field over an algebraically closed field.

Lerch’s Theorems over Function Fields

In this work, we state and prove Lerch’s theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. 1. Results The Fermat’s little theorem states that if p is a prime and a is an integer not divisible by p, then ap−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = ap−1 − 1 p , which is an integer. This q...

Arithmetic Duality Theorems Second Edition

Addendum/Erratum for Arithmetic Duality Theorems

Then H need not be divisible, i.e., we need not have pH D H . We only know H is contained in pG. Moreover, for x in H , there need not exist an infinite sequence y1;y2;y3:::: such that py2 D y1, py3 D y2; ::. We only know that there exist arbitrarily long finite such sequences. There does exist a unique maximal divisible subgroup D of M , and D is a subgroup of H . Moreover, M DD ̊N where N is a...

Duality Theorems in Galois Cohomology over Number Fields

the direct limit taken over all finite Galois extensions K of k in which the integral closure Y of X is unramified over X, where GKJk denotes the Galois group of such an extension, and where GY denotes the group of points of G with coordinates in Y. For example, if X=k, our notation coincides with that of [10]. For any X, the group H(X,C) is the r-th cohomology group of the profinite group Gjr=...