## David Goss

This home page contains information that I believe will be of interest to mathematicians, in general, and number theorists, in particular. It contains links to various preprints, papers, and books that I have written or edited. (In particular, there is a sample chapter from, my book "Basic Structures of Function Field Arithmetic," [cover.jpg] which was published on October 17, 1996 as well as information on the soft-covered "study" edition published on November 18, 1997.) Finally it contains a link to files on mathematical writing. It should be considered as being in perpetual construction.

January 28, 2006: In a paper here.dvi , here.pdf , written honor of the 300-th birthday of Leonhard Euler, we considered L-series in characteristic p from an "Eulerian" perspective (see the paper by Ayoub in the references). In particular, we discuss old calculations of Dinesh Thakur and current calculations of Javier Diaz-Vargas of "non-classical" trivial zeroes: In algebraic number theory, the order of a trivial zero is easily computed via the functional equation. In characteristic p, as Thakur found, there are many instances where the order is "non-classical"; i.e., higher than expected. It is remarkable that in the examples computed, these non-classical trivial zeroes occur at -j where the sum of the p-adic digits of j is bounded. We present a general conjecture along these lines and discuss how it fits with previous work on the zeroes thereby giving some "justification" to the conjecture in an Eulerian sense. The solution to this conjecture may very well require the correct "functional equation" in the characteristic p theory.

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 ALL associated partial $L$-series. Our proof uses the logarithmic growth and certain deep estimates of Amice. An interesting point of the proof is the way that the analytic theory of L-series at all(!) places of base field k is needed to establish the result. (This preprint has appeared in The Journal of Number Theory, {\bf 110} (2005) 83-113.) For a copy, go here.dvi, here.ps, or here.pdf.

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.

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.

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.

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.