# Abstracts

 On the direct problem in differential Galois theory Lucia Di Vizio (Université Versailles-St Quentin) I will describe an algorithm (implemented in Maple) that produces a system of generators of the Lie algebra of an aboslutely irreducible linear differential equation over the rational functions with complex coefficients. This is a joint work with M. Barkatou, T. Cuzeau et J.-A. Weil.
 On exponents of class groups in towers of number Fields Farshid Hajir (University of Massachusetts, Amherst) The ideal class group is simultaneously one of the most basic and one of the most mysterious objects in number theory. In particular, how the class group varies in towers of number fields presents a host of interesting questions. More than 50 years ago, Iwasawa gave a beautiful formula for the growth of the size of the $p$-part of class groups in abelian $p$-adic analytic extensions. Such extensions are always wildly ramified. We know much less about how class groups behave in unramified towers. In this lecture, I will talk about a joint work with Christian Maire in which we construct unramified $p$-towers in which the average exponent of $p$-class groups remains bounded.
 Hardy-Littlewood numbers and Bessel's functions Alessandro Languasco (Università di Padova) We describe a joint result with A. Zaccagnini about an explicit formula involving non-trivial zeros of the Riemann zeta-function for the Cesàro averaged number of representations of a non-square integer as a sum of a prime and a square. We'll see that Bessel's functions of complex order and real argument play a role in such an explicit formula.
 Analytic Lie extensions of number fields with cyclic fixed points and tame ramification (joint work with Farshid Hajir) Christian Maire (Université Franche-Comté) In this lecture, I will present an extension of the strategy of Boston from the 90's concerning the tame version of the Fontaine-Mazur conjecture. More precisely, we are interested in using group-theoretical information to derive consequences for tamely ramified Galois representations, specially for the groups Sl$(n,\mathbf{Z}_p)$.
 Counting constrained almost primes Pieter Moree (Max-Planck-Institut für Mathematik) A natural number is called $k$-almost prime if it has exactly $k$ prime factors, counted with multiplicity. In my talk I consider the asymptotic counting of such numbers (mostly with $k\le 3$) with additional constraints put on the prime factors. E.g., if we take $k=2$ and write $n=pq$ with $p$ and $q$ primes of similar size we obtain the so called RSA-integers that play an important role in cryptography. We also consider some examples with $k=3$ inspired by cryptography and the study of coefficients of cyclotomic polynomials. The talk is based on three papers with as coauthors Decker (2008), Camburu, Ciolan, Luca and Shparlinski (2016) and Eddin (2017).
 A rigidity theorem for translates of uniformly convergent Dirichlet series (joint work with M. Righetti) Alberto Perelli(Università di Genova) It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<\sigma<1$; this is certainly not true for $\sigma>1$. Answering a question of Bombieri and Ghosh, we give a simple characterization of the analytic functions approximable by translates of $L$-functions in the half-plane of absolute convergence. Actually, this is a special case of a general rigidity theorem for translates of Dirichlet series in the half-plane of uniform convergence. Our results are closely related to Bohr's equivalence theorem.
 Arguments of exponential sums René Schoof (Università di Roma "Tor Vergata") We prove some special cases of a conjecture involving arguments of certain exponential sums.
 Primitivity properties of points on elliptic curves Peter Stevenhagen(Universiteit Leiden) Given an elliptic curve $E$ over a number field $K$ and a point $A$ in $E(K)$, one may ask, following Lang and Trotter, for how many primes $p$ of $K$ the point $A$ is $primitive$ at $p$, i.e., generates the point group of the reduced curve ($E$ mod $p$). We present a characterization, in terms of the Galois representation of $E$, of the phenomenon of ''never-primitive points''.
 Hypergeometric motives: Hodge numbers and supercongruences Fernando Rodriguez Villegas (ICTP) I will discuss a conjectural connection between higher order congruences for certain finite hypergeometric sums and the Hodge numbers of the associated hypergeometric motive. The congruences in question extend those of Dwork for the unit root of the corresponding local L-series.
 Diophantine approximation of power series Michel Waldschmidt (Université Pierre et Marie Curie) We give an introduction to the theory of Diophantine approximation of power series, starting with continued fractions and culminating with parametric geometry of numbers.