The Skolem-Mahler-Lech
theorem is classical result concerning the vanishing of linear
recurrences over fields of characteristic zero. In this talk, I
will discuss some results inspired by analogous number theoretical
questions over fields of positive characteristic. I will
especially focus on some decision problems related to these
questions. This is a joint work with Jason Bell.

Francesco Amoroso

Dans cet exposé, je
donne une variante effective presque optimale du lemme matriciel
de Masser, qui consiste à minorer la hauteur d'une
variété abélienne en fonction de ses
réseaux des périodes. Ce résultat est
utilisé par Gaudron et Rémond pour établir de
nouvelles versions effectives du fameux théorème des
périodes (de Masser et Wüstholz).

We present several results
about the spectrum of singularities and the fractal dimension of
the graph of some Fourier series having polynomial frequencies. We
also discuss the case of fractional integrals of modular forms.
Both cases include variants on the so-called "Riemann example".

**Chieh-Yu Chang**, *On multiple polylogarithms and multiple
zeta values in positive characteristic,*

In this talk, we consider the Carlitz multiple polylogarithms (CMPLs) at algebraic points. We show that they form a graded algebra defined over the base rational function field. We further show that any multiple zeta value defined by Thakur can be expressed as a linear combination of CMPLs at algebraic points, which is a generalization of the work of Anderson-Thakur on the depth one case. As a conseqence, we obtain a function field of Goncharov's conjecture for MZVs.

In this talk, we consider the Carlitz multiple polylogarithms (CMPLs) at algebraic points. We show that they form a graded algebra defined over the base rational function field. We further show that any multiple zeta value defined by Thakur can be expressed as a linear combination of CMPLs at algebraic points, which is a generalization of the work of Anderson-Thakur on the depth one case. As a conseqence, we obtain a function field of Goncharov's conjecture for MZVs.

Sara Checcoli

Let G be a torus or an
abelian variety and let V be a proper subvariety of G. A central
problem in diophantine geometry is to understand when a geometric
assumption on V is equivalent to the non-density of certain
'special' subsets of V. The Manin-Mumford, Mordell-Lang and
Zilber-Pink conjectures are of this nature. Bombieri, Masser and
Zannier gave, in the toric case, a new approach to this kind of
problems. In particular, they introduced the notion of
V-torsion anomalous intersection: it is an intersection, having
components of dimension bigger than expected, between V and a
translate of a subgroup by a torsion point. They propose some
non-density and finiteness conjectures for this kind of
intersections. After an introduction on the subject, I will
present some results obtained with F. Veneziano and E. Viada when
G is a product of elliptic curves with complex multiplication.

Let M(r) be the set of all real numbers that are approximable by the rationals at a rate at least some given r >= 2. More than eighty years ago, Jarnik and, independently, Besicovitch established that the Hausdorff dimension of M(r) is equal to 2/r. We consider the further question of the size of the intersection of M(r) with Ahlfors regular compact subsets of the interval [0,1]. In particular, we propose a conjecture for the exact value of the dimension of M(r) intersected with the middle-third Cantor set. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. This study relies on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points. This is a joint work with Yann Bugeaud (Strasbourg).

Étant donné une période w d'une variété abélienne A (définie sur un corps de nombres k), notons A_w la plus petite sous-variété abélienne de A dont l'espace tangent à l'origine contient w. Un théorème des périodes donne une majoration du degré de A_w, degré relatif à une polarisation sur A. Les premières bornes ont été obtenues par Masser et Wüstholz dans les années 90. Ces énoncés permettent par exemple d'estimer le degré minimal d'une isogénie entre deux variétés abéliennes isogènes. Dans cet exposé, nous présenterons la borne du degré de A_w obtenue en collaboration avec Gaël Rémond ainsi que les principales idées de la démonstration.

We discuss various results regarding the variation of the canonical height in families of rational maps and their connection to the problem of "unlikely intersections" in arithmetic dynamics.

Samuel Le Fourn

As explained in previous talks of this conference, a combination of Mazur's method, isogeny theorems and Runge's method provides a result of nonexistence of rational points on the modular curves Xsplit(p). In this talk, after recalling the guiding principles of these techniques, I will use them to solve a different question (about Q-curves), as an excuse to describe more generally when they can expectedly be applied and successful.

For E/K an elliptic curve without complex multiplication we bound the index of the adelic image of Gal(Kbarre/K) in GL_2(Z^), the representation being given by the action on all the Tate modules of E at the various primes. The bound is effective and only depends on [K: Q] and on the (stable) Faltings height of E.

Pierre Parent

We describe a method allowing to prove that a certain family of modular curves of level p, for very large prime p, have no rational points but an explicit (and very short) list, namely complex-multiplication points and cusps. The basic idea consists in proving that the height of a rational point is stuck between a lower and an upper bound, which become uncompatible for large enough p. The lower bound is a consequence of the isogeny theorem, whose proof will have been explained in the previous lectures. The upper bound comes from an old diophantine technique called "Runge's method" (which shall also be discussed further in the talk of Samuel Le Fourn). The above is joint work with Yuri Bilu. The question of small primes will be tackled in Marusia Rebolledo's lecture. The final output applies to the so-called "Serre's uniformity problem".

In this talk we describe some results obtained in collaboration with B. Anglès and F. Tavares Ribeiro. We propose to study Drinfeld modules over Tate algebras and associate to them certain $L$-series values. These $L$-series values have a double status of numbers and functions. As numbers, they satisfy a variant of the class number formula. As functions they satisfy certain functional identities. As applications of our investigations, we will mention a refinement of Anderson's log-algebraicity theorem and a refinement of Herbrand-Ribet-Taelman theorem.

In my talk, I will speak
about the local behaviour of certain Fourier series, which are
related to modular forms. Since modular forms of even weight can
be expressed in terms of Eisenstein series, I will first focus
on the Fourier series arising from Eisenstein series and
generalise the results in the last part of the talk. In our
analysis we apply wavelets methods proposed by Jaffard in 1996
in the study of the Riemann series and we use the modularity
(and quasimodularity) of Eisenstein series. We find that the
Hölder regularity exponent at an irrational x is related to
the continued fraction expansion of x, in a very precise way.

**Marusia Rebolledo**,
*Rational points on X_0^+(p^r) for small primes p*

This talk is the second of a series of
lectures about the use of diophantine methods in the study of
rational points on modular curves. In the first talk, Pierre
Parent will explain how to prove, conjugating an isogeny
theorem and a method due to Runge, that for p great enough
(this is explicit, roughly for p>10^11) the modular curve
X_0^+(p^r) has no rational points other than cusps and CM
points. I will tackle with the "small" primes (p<10^11).
The techniques involved are a mix of a variant of Mazur's
method, a Gross formula for special values of L-functions and
algorithmic.

Gaël Rémond

Nous montrons comment le
théorème des périodes présenté
dans l'exposé d'Éric Gaudron permet d'établir
diverses estimations explicites pour la géométrie
des variétés abéliennes sur les corps de
nombres. En particulier, nous démontrons qu’il existe un
faisceau inversible ample et symétrique sur une telle
variété abélienne A dont le degré est
borné par une expression explicite qui dépend
seulement de la dimension de A, de sa hauteur de Faltings et du
degré du corps de nombres. Nous donnons aussi des versions
améliorées et explicites des théorèmes
d'isogénie de Masser et Wüstholz, notamment dans le
cas elliptique utilisé par Bilu, Parent et Rebolledo en
direction du problème d'uniformité de Serre. Tous
les résultats de l'exposé sont issus d'un travail en
commun avec Éric Gaudron.

**Jean-Pierre Serre**, * How to prove that Galois groups are
"large"**,*

The Galois groups of the title are those which are associated with elliptic curves over number fields; I shall explain the methods which were introduced in the 1960's in order to prove that they are large, and the questions about them which are still open fifty years later.

**Stéphane Seuret**, *p-exponents and p-multifractal
spectrum of some lacunary Fourier series,*

We study the Fourier series R_s(x) = sum_{n >= 1} sin (pi n^2 x)/n^s when 0<s<1. In this range of parameters, It is easily seen that R_s does not converge everywhere. We prove that the convergence of R_s(x) depends on Diophantine properties of x. Then, at each point of convergence x, we compute the L^2-pointwise exponent of R_s, and we deduce its L^2-multifractal spectrum.

**Martin Sombra**, *Height of varieties over finitely
generated fields**,*

Moriwaki defines the height of a cycle over an arbitrary finitely generated extensions of Q as an arithmetic intersection number in the sense of Gillet and Soulé. We show that this height can be written in local terms. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to some arithmetic intersection numbers of non toric varieties (joint work with J. I. Burgos and P. Philippon).

Sanju Velani, *Well on fibres and Bad on curves*,

Khintchine’s classical theorem provides an elegant zero-one criterion for the set of well approximable points in the theory of simultaneous Diophantine approximation. I will start by discussing a recent strengthening of this theorem. I will then move onto discussing the problem of intersecting simultaneous badly approximable sets with planar curves.

**Martin Widmer**, *Diophantine
approximations, flows on homogeneous spaces and counting*

The Galois groups of the title are those which are associated with elliptic curves over number fields; I shall explain the methods which were introduced in the 1960's in order to prove that they are large, and the questions about them which are still open fifty years later.

We study the Fourier series R_s(x) = sum_{n >= 1} sin (pi n^2 x)/n^s when 0<s<1. In this range of parameters, It is easily seen that R_s does not converge everywhere. We prove that the convergence of R_s(x) depends on Diophantine properties of x. Then, at each point of convergence x, we compute the L^2-pointwise exponent of R_s, and we deduce its L^2-multifractal spectrum.

Moriwaki defines the height of a cycle over an arbitrary finitely generated extensions of Q as an arithmetic intersection number in the sense of Gillet and Soulé. We show that this height can be written in local terms. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to some arithmetic intersection numbers of non toric varieties (joint work with J. I. Burgos and P. Philippon).

Sanju Velani

Khintchine’s classical theorem provides an elegant zero-one criterion for the set of well approximable points in the theory of simultaneous Diophantine approximation. I will start by discussing a recent strengthening of this theorem. I will then move onto discussing the problem of intersecting simultaneous badly approximable sets with planar curves.