Automorphisms of non-split Cartan modular curves|
Valerio Dose (Università di Roma Tor Vergata)
The interest in non-split Cartan modular curves comes mainly from their relation to Serre's uniformity conjecture,
an important statement about Galois representations attached to elliptic curves.
In this talk, we will present recent results on the automorphism group of non-split Cartan modular curves of prime level.
The distribution of class groups of imaginary quadratic fields|
Nathan Jones (University of Illinois at Chicago)
Which abelian groups occur as the class group of some imaginary quadratic field? Inspecting tables of M. Watkins on imaginary quadratic fields
of class number up to 100, one finds that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance (Z/3Z)3 does not).
In this talk, I will combine heuristics of Cohen-Lenstra together with a refinement of a conjecture of Soundararajan to make precise predictions about the
asymptotic distribution of imaginary quadratic class groups, partially addressing the above question. I will also present some numerical evidence of the
resulting conjectures. This is based on joint work with S. Holmin, P. Kurlberg, C. Mc.Leman, and K. Petersen.
Automata and number theory|
Christian Mauduit (Université d'Aix-Marseille)
The aim of this talk is to give a survey on recent results concerning the combinatorial, arithmetical
and statistical properties of sequences of symbols and sequences of integers generated by finite automata.
We will illustrate our talk by showing some classical constructions, including the Thue-Morse sequence,
the Rudin-Shapiro sequence and the Cantor sequence.
New Results on the Class Number One Problem for Function Fields|
Pietro Mercuri (Università di Roma Tor Vergata)
In this talk we give a complete classification of function fields with class number one and positive genus
and we explain the role of pointless curves in this result. This is a joint work with Claudio Stirpe.
Forbidden integer ratios of consecutive power sums|
Pieter Moree (Max-Plank Institute fur Mathematics)
Sk (m):=1k+2k+ . . . . +(m-1)k
denote a power sum. In 2011 Bernd
the conjecture that for k≥ 2 and m≥ 4> the ratio
r:=Sk(m+1)/Sk(m) of two consecutive power sums is never an
integer. In case r=2 this is equivalent with the Erdos-Moser conjecture that
Sk(m)=mk has no non-trivial solutions.
I will discuss criteria allowing one to exclude many integer values of r as
possible ratio (joint work with Ioulia Baoulina).
On some extensions of the Ailon-Rudnick Theorem|
Alina Ostafe(University of New South Wales)
Digital properties of prime numbers|
Joël Rivat (Université d'Aix-Marseille)
We will give an introduction and a survey of some recent
results on the distribution of the digits of prime numbers
obtained in collaboration with Christian Mauduit,
focusing in particular our attention on the Thue-Morse and Rudin-Shapiro
sequences along prime numbers.
Adelic points of ellipitc curves|
Peter Stevenhagen (Leiden Universiteit)
Parametric Pell equations|
Leonardo Zapponi (Université Pierre et Marie Curie)