Journal de Théorie des Nombres de Bordeaux

Rudolph Perkins

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Alexander Gorodnik

Shirali Kadyrov

Diophantine approximation with restricted numerators and denominators on semisimple groups

Denis Koleda

On the density function of the distribution of real algebraic numbers

Frédéric Chapoton

Jiang Zeng

Nombres de $q$-Bernoulli–Carlitz et fractions continues

Ambrus Pál

Federico Pellarin

Lenny Taelman

Ambrus PÁL, Federico PELLARIN et Lenny TAELMAN
Foreword
Tome 29, no 3 (2017), p. i-iii.
<http://jtnb.cedram.org/item?id=JTNB_2017__29_3_1_0>
© Société Arithmétique de Bordeaux, 2017, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Foreword
The reader of this volume holds in his hands the proceedings of the
workshop on function fields, zeta functions and Drinfeld modular forms
held from 22th to 26th June, 2015, at Imperial College London, honoring
the 61st birthday of David Goss. David, who was also a close personal friend
of the organisers, passed away tragically on the 4th of April, 2017, so we
would like to dedicate this volume to his memory, and write a few words
about the significance of his work in function field arithmetic.
David received his PhD in 1977 from Harvard under the supervision of
Barry Mazur. He held positions at Princeton, Berkeley, and Brandeis before
joining Ohio State University in 1982, where he retired in 2013 and became
Professor Emeritus. David was generous with his time and knowledge and
was full of enthusiasm and support for his colleagues, especially junior re-
searchers in number theory. He was considered by many as a mentor and
a friend, and his broad and deep knowledge as well as his kindness and
warmth will be missed by many.
David’s research was in algebraic number theory and algebraic geometry,
where he made extensive contributions to the arithmetic theory of function
fields, publishing some 40 academic papers. He was also the author of “Basic
Structures of Function Field Arithmetic”, which is one of the most popular
and accessible references in this topic. Starting with his Harvard thesis,
David introduced a theory of modular forms for (k, ∞), with k a global
field of positive characteristic and ∞ a fixed place of k. In 1979, while
at Princeton, he accomplished another important advance, by introducing
zeta and L functions for (k, ∞) (now called Goss’ zeta and L-functions). In
this seminal work, he also introduced the famous character group S∞ and
the zeta-phenomenology of analytic continuation, the negative values, the
trivial zeroes, and other highly inspirational structures. He later continued
developing his theory of zeta and L-functions in the path of Euler and
Riemann, and pioneered this vast area, such as the analogue of the Riemann
hypothesis, the functional equation, the Gamma factors.
The main stay of the conference was one of the primary interests of
David, namely the study of special values of L-functions of characteristic p
ii
values. The broader subject of special values of L-functions has significant
connections to other important areas of mathematics such as the study of
Galois representations, algebraic cycles, motives, K-theory, transcendence
theory, modular forms and Iwasawa theory. In the last few years, the study
of special values of L-functions of objects defined over positive characteristic
fields, such as algebraic varieties and t-motives, experienced particularly
rapid growth. Some of these advances used powerful cohomological trace
formulas available in this setting; however other methods are more exotic,
and hence mysterious, including deformations of Drinfeld modular forms.
These in turn introduced a host of new problems about the latter objects
and spurred a great deal of activity. As mentioned above, characteristic
p-valued L-functions attached to Drinfeld modules and t-motives, as well
as Drinfeld modular forms, were first introduced by Goss. These objects
have familiar flavour if compared to their counterparts over number fields,
that is, they exhibit similar analytic behaviour and arithmetic behaviour
of their special values; at least at first sight. Many of the basic properties
of these L-functions are now well understood. In particular Taguchi and
Wan proved, using analytic techniques introduced by Dwork, that these
L-functions have entire continuation to S∞. In another major development
Böckle and Pink found a cohomological proof of this analytic behaviour, via
a trace formula. Of course we would like to have an interpretation of the
special values of these L-functions in terms of global arithmetic objects,
a main theme in number theory, which is perhaps the most fascinating,
and enigmatic, and important progress has been made in the last couple
of years, notably including variants of the class number formula. And so
much remains to be done!
In the present volume, the reader will find many faces of David’s
arithmetic dream. The theory of modular forms with values in complete
fields of positive characteristic, including v-adic interpolation, higher rank
modular forms, is considered in the contributions of Papanikolas and Zeng,
Breuer and Basson, Gekeler, Perkins. Aspects of zeta and L-functions, as
well as their values, but also more classical questions of algebraic number
theory or transcendence, are investigated in the works of Rosen, Anglès,
Ngo Dac and Tavares Ribeiro, as well as Boeckle and Thakur, and Pellarin.
Studies with more geometric flavour, related to characteristic p Galois rep-
resentations, crystals, rigid cohomology, are tackled by Lazda, Heuer and
Pál(1). An extension of zeta values (introduced by Thakur) is given by mul-
tiple zeta values which are considered, in the volume, through analogues
in positive characteristic which appear as a not fully understood alter ego
of the real ones, by Chen, Thakur, Pellarin, and Chang and Mishiba. The
volume begins with a paper of David, containing some of his views on
(1)Pál’s article is currently under review and is not included in this issue.
Foreword iii
measures, distributions, digit calculus, Riemann’s hypothesis, and his “crazy
group S(q),” as he liked to call it. Some of the structures he introduced, like
this one, remain mysterious. May this volume be an inspiration, especially
to younger researchers, to pursue the challenges that he put forward.
Ambrus Pál, Federico Pellarin, Lenny Taelman.
6th December 2017
Note of the editor As for all proceedings published by JTNB, the articles
in this issue followed the usual editorial process and are referred as normal
submissions.


Foreword

David Goss

Digit permutations revisited

Mark W. Coffey

James L. Hindmarsh

Matthew C. Lettington

John D. Pryce

On Higher-Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents

Akinari Hoshi

Complete solutions to a family of Thue equations of degree 12

Dirk Basson

Florian Breuer

On certain Drinfeld modular forms of higher rank

Davide Lombardo

Pink-type results for general subgroups of $\operatorname{SL}_2(\mathbb{Z}_\ell )^n$

Not a member yet

Twisting eigensystems of Drinfeld Hecke eigenforms by characters

Diophantine approximation with restricted numerators and denominators on semisimple groups

On the density function of the distribution of real algebraic numbers

Nombres de $q$ -Bernoulli–Carlitz et fractions continues

Foreword

Digit permutations revisited

On Higher-Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents

Complete solutions to a family of Thue equations of degree 12

On certain Drinfeld modular forms of higher rank

Pink-type results for general subgroups of $\operatorname{SL}_2(\mathbb{Z}_\ell )^n$

1,235

1,351