The first mini symposium of the

Roman Number Theory Association

Università Europea di Roma

7 Maggio 2015



Titles and abstracts


Title and Abstracts

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 Kellner formulated 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)