Quite recently, Gebhard Boeckle, wrote an ETH habilitation thesis in which he used the theory of "crystals" for function fields to associate Galois representations to Hecke cuspidal and double cuspidal eigenforms in finite characteristic. (A copy of this thesis is available the site linked to Boeckle below.) This is major breakthrough in the characteristic p theory. It is also a complex work which uses almost all the machinery invented so far! Boeckle's Galois representations are abelian (in contrast to what happens classically, although even classical Dirichlet characters arise from modular forms of fractional weight). In the simplest case where our base algebra is the ring of polynomials over a finite field, it then makes sense to ask if rank 1 Drinfeld modules can be modular in that their associated Galois representations arise from modular forms. In a preprint, (what follows is the final, edited, version) here.dvi , here.pdf , here.ps , we present an introduction to Boeckle's work with an emphasis on modularity in both the classical (elliptic curves over Q AND over a function field) and characteristic p theories. This preprint is published in the Journal of the Ramanujan Math. Soc. {\bf 17} No. 4 (2002) 221-260.

In Mathematische Annalen 323 (2002) 737-795, Gebhard Boeckle shows how to analytically continue the L-series associated to "tau-sheaves." Such sheaves arise naturally from Drinfeld modules as well as general A-modules (in the sense of G. Anderson) and have become the basic structure of characteristic p arithmetic. Boeckle's proof uses essentially the logarithmic growth of the degrees of the "special polynomials" associated to these L-series. In a new paper, we show how to use non-Archimedean integration to analytically continue such L-series as well as

In a 1999 preprint we explained how to formulate the characteristic p "Generalized Riemann Hypothesis," as well as the characteristic p version of the "Generalized Simplicity Conjecture," in terms of absolute values. A corresponding reformulation of the classical statements is also given. The similarities between the "absolute value conjectures" in the two theories (i.e., classical and characteristic p arithmetic) are very strong and quite surprising. (See also the next two paragraphs.) For a copy of this preprint, go here.dvi, here.ps, or here.pdf. The published version of this paper is Journal of Number Theory vol. 82, no.2 (June, 2000) 299-322. The on-line version can be accessed at here. (See below for a site dedicated to current papers on the Riemann Hypothesis.)

The Riemann Hypothesis in characteristic p is based on the work of D.Wan, D.Thakur and J.Diaz-Vargas, B.Poonen, and J.Sheats (exact references given in the above files) for the zeta function of F_r[T]. The zeta function is defined for (x,y) in S_\infty where x lies in a finite characteristic complete field and y is a p-adic integer. Thus one has a 1-parameter family of power series and an associated 1-parameter family of Newton polygons. One can therefore say more precisely that Wan, Diaz-Vargas, etc., compute the family of Newton polygons associated to this zeta-function. This family turns out to have segments with horizontal length identically 1, which establishes that the zeroes are both simple (i.e., of order 1) and in Fr((1/T)). In seeking to phrase this ``Rh'' for as large a class of characteristic p functions as possible, it was realized that one would have to allow for finitely many anomalous zeroes for each y. Now for y a negative integer one only has finitely many trivial zeroes; thus we simply ignored these trivial zeroes (and so the infinite primes) in formulating these conjectures.

Since the publication of the above manuscript, it was realized that
this assumption is in error; that is, the trivial zeroes, and so the
infinite primes, must be taken into account. What happens is that
the topology on S_\infty allows us to use zeroes close to trivial
zeroes (``near-trivial zeroes'') to inductively construct counter-examples.
This is contained in
``The impact of the infinite primes on the Riemann
hypothesis for characteristic p valued L-series,''
which appears in the book "Algebra, Arithmetic and Geometry with
Applications",
(Eds: Christensen et al) Springer (2004) 357-380. For
the preprint see
**here.dvi,
here.ps**
or **
here.pdf,**. In this paper we also begin to come to
grips with the implications of these counter-examples.
We explain how trivial and near-trivial
zeroes should
arise in general, and how they might ultimately be handled
via Hensel's Lemma (whereas classically one uses Gamma-functions).
We break up all the zeroes into two classes,
the near-trivial zeroes and the ``critical zeroes'' (= all other
zeroes). But then there is a remarkable surprise: All zeroes
computed by Wan et al are near-trivial (this is established
when r=p and should be true in general). Thus, while it is reasonable to
believe that these critical zeroes will exist in abundance, up to now
we have had very little experience with them.

Follow this link
**
style** for files on mathematical writing.

The publisher for the "Basic Structures" volume is
**Springer ** and the Springer
series which contains the book is **
Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 35. **

**Abstract for volume 35:**
The arithmetic of function fields over finite fields has been an
area of extensive research for about two decades with many
tantalizing results having recently been obtained. This book
offers a self-contained introduction to basic concepts such as
Drinfeld modules, T-modules, shtukas, exponentiation of ideals, and
characteristic p L-functions and Gamma-functions in all their
various manifestations. Insight from classical number theory,
differential equations/algebra and algebraic geometry is presented
whenever needed. Interesting unsolved problems are posed to the
reader and a comprehensive list of references is included. All of
this is presented to give the reader the basic tools to begin
doing active research into this rapidly expanding area.

The volume is ISBN 3-540-61087-1 and contains 415 references.

Below is the .dvi file of the table of contents.

**Table of Contents of Vol. 35**- Follow this link
**Springer Germany**for the homepage of Volume 35 on www.Springer.de. - Follow
this link
**Springer New York**for the homepage of Volume 35 on www.Springer-ny.com.

The ** soft-covered study edition** is priced
at $59.95 and DM 108,- . It has 453 references and some minor corrections
have been made. The ISBN number is 3-540-63541-6.
In conjunction with Springer, I am very happy to be able to offer both
a list of these corrections as well as the expanded references below.
** In addition, still in conjunction with Springer, I am delighted to
offer the .dvi, .ps, and .pdf files of Section 3 "The Carlitz Module" as a
sample of the book.**

**For the .dvi file of Section Three.****For the .ps file of Section Three.****For the .pdf file of Section Three.****For the .dvi file of the references.****For the .ps file of the references.****For the .pdf file of the references.**-
**For the list of corrections.**

In June, 1991, there was a conference at Ohio State
on the arithmetic of function fields related to Drinfeld modules. The
proceedings were published by de Gruyter in 1992
as ** "The Arithmetic of Function Fields,"** which is the
second volume of de Gruyter's ** Ohio State University Mathematical
Research Institute Publications.** It was edited by D. Goss, D.R. Hayes
and M.I. Rosen and is ISBN 3-11-013171-4.

Below is the .dvi file of the table of contents of these proceedings.

To find out more about this volume from de Gruyter, follow the next link and do a search under "Ohio" in the series box.

The following are links to preprints of some other mathematicians interested in the arithmetic of function fields. If you know of other links that perhaps should be added to this list, please contact me at the address given below.

- Greg Anderson
- Gebhard Boeckle
- Gunther Cornelissen
- Henri Darmon
- Ernst-Ulrich Gekeler and his school
- Dinesh Thakur
**Doug Ulmer**

For a general list of ** number theorists,**
go **here.**
For general list of **number theory preprints,** go**
here ** or **
here . ** For a collection of
current papers about the ** Riemann Hypothesis **
go ** here
** .

**
The address of the Department of Mathematics at The Ohio State University
is 100 Mathematics Building,
231 W. 18-th Avenue, Columbus Ohio 43210-1174.
I can be reached by email at
goss@math.ohio-state.edu,
by phone at
614-688-5773, and by fax at 614-292-1479. All correspondence concerning
the Journal of Number Theory should be sent to
jnt@math.ohio-state.edu.
TO SUBMIT TO THE JOURNAL OF NUMBER THEORY, GO
HERE.
**

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