# Cohomological Theory of Crystals over Function Fields

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*Gebhard Böckle; Richard Pink*

A publication of the European Mathematical Society

This book develops a new cohomological theory for schemes in positive
characteristic \(p\) and it applies this theory to give a purely
algebraic proof of a conjecture of Goss on the rationality of certain
\(L\)-functions arising in the arithmetic of function fields. These
\(L\)-functions are power series over a certain ring \(A\), associated to
any family of Drinfeld \(A\)-modules or, more generally, of
\(A\)-motives on a variety of finite type over the finite field
\(\mathbb{F}_p\). By analogy to the Weil conjecture, Goss conjectured
that these \(L\)-functions are in fact rational functions. In 1996
Taguchi and Wan gave a first proof of Goss's conjecture by analytic methods
à la Dwork.

The present text introduces \(A\)-crystals, which can be viewed as
generalizations of families of \(A\)-motives, and studies their
cohomology. While \(A\)-crystals are defined in terms of coherent
sheaves together with a Frobenius map, in many ways they actually behave like
constructible ètale sheaves. A central result is a Lefschetz trace
formula for \(L\)-functions of \(A\)-crystals, from which the
rationality of these \(L\)-functions is immediate. Beyond its
application to Goss's \(L\)-functions, the theory of
\(A\)-crystals is closely related to the work of Emerton and Kisin on
unit root \(F\)-crystals, and it is essential in an Eichler –
Shimura type isomorphism for Drinfeld modular forms as constructed by the first
author.

The book is intended for researchers and advanced graduate students
interested in the arithmetic of function fields and/or cohomology
theories for varieties in positive characteristic. It assumes a good
working knowledge in algebraic geometry as well as familiarity with homological
algebra and derived categories, as provided by standard textbooks. Beyond that
the presentation is largely self contained.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic.