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About this book

This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.

Table of Contents


Cohomological Theory of Crystals over Function Fields and Applications

This lecture series introduces in the first part a cohomological theory for varieties in positive characteristic with finitely generated rings of this characteristic as coefficients developed jointly with Richard Pink. In the second part various applications are given.
Gebhard Böckle

On Geometric Iwasawa Theory and Special Values of Zeta Functions

Having succumbed to the requests of the organisers of the Research Programme on Function Field Arithmetic that was held in 2010 at the CRM in Barcelona, we present here a survey of some recent results concerning certain aspects of the Iwasawa theory of varieties over finite fields.
David Burns, Fabien Trihan

The Ongoing Binomial Revolution

The Binomial Theorem has played a crucial role in the development of mathematics, algebraic or analytic, pure or applied. It was very important in the development of calculus, in a variety of ways, and has certainly been as important in the development of number theory. It plays a dominant role in function field arithmetic. In fact, it almost appears as if function field arithmetic (and a large chunk of arithmetic in general) is but a commentary on this amazing result. In turn, function field arithmetic has recently returned the favor by shedding new light on the Binomial Theorem. It is our purpose here to recall the history of the Binomial Theorem, with an eye on applications in characteristic p, and finish by discussing these new results.
David Goss

Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields

In the advanced course given at Centre de Recerca Matem`atica, consisting of twelve hour lectures from 22 February to 5 March 2010, we described results and discussed some open problems regarding the gamma and zeta functions in the function field context. The first four lectures of these notes, dealing with gamma, roughly correspond to the first four lectures of one and half hour each, and the last three lectures, dealing with zeta, cover the last three two-hour lectures. Typically, in each part, we first discuss elementary techniques, then easier motivating examples with Drinfeld modules in detail, and then outline general results with higher-dimensional t-motives. Lecture 4 is independent of Lecture 3, whereas the last part (last three lectures) is mostly independent of the first part, except that the last two lectures depend on Lecture 3. At the end, we include a guide to the relevant literature.
Dinesh S. Thakur

Curves and Jacobians over Function Fields

These notes originated in a 12-hour course of lectures given at the Centre de Recerca Matem`atica in February 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell–Weil rank.
Douglas Ulmer


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