We will have a series of seminars in October and November 2024. The talks will take place in Aula INdAM at 12.
2024.10.4. Evelina Viada (University of Göttingen)
On the torsion and rational points of some families of curves
I will present an overview of some explicit methods in the context of the Manin-Mumford and Mordell Conjecture. I will give several specific examples (done with R. Pengo or F. Veneziano) of families of curves whose rational and torsion points can be explicitly determined. I intend to explain some of the techniques that we use, among other there are some results in Galois representation, in the geometry of numbers and in arithmetic geometry. I will avoid the details trying to keep the talk understandable to a large audience.
2024.10.11. Maria Valentino (Università della Calabria)
Drinfeld quasi-modular forms of higher level
It is well known that the space of modular forms is not stable under differentiation. This is why quasi-modular forms have been introduced. Also Drinfeld modular forms, that are the positive characteristic counterpart of classical modular forms, are not stable under differentiation. In 2008 V. Bosser and F. Pellarin introduced Drinfeld quasi-modular forms of level 1. In this talk we will present some new and recent advances on Drinfeld quasi-modular forms of higher level.
2024.10.25. Andrea Bandini (Università di Pisa)
Iwasawa theory for l-parts in pro-p-extensions and a theorem of Sinnott
Iwasawa theory studies arithmetically significant modules (e.g. class groups and Selmer groups) associated with pro-p-extensions K/k of global fields (p a prime). It usually focuses on p-parts of such modules and few results are known on l-parts (l different from p another prime), mainly obtained by means of analytic methods. We present an algebraic approach to study l-parts as modules over the algebra Z_l[[Gal(K/k)]], providing structure theorems, characteristic ideals, orders, Z_l-ranks and so on. In the case of class groups, such modules naturally verify a theorem of Sinnott on the p-adic limit of their orders (or their Z_l-ranks when they are not finite). We show that this holds for more general modules and (if time permits) conclude with a (tentative) formulation of a Main Conjecture for this setting. This is joint work with Ignazio Longhi (Torino).
2024.10.29. Francesco Battistoni (Università Cattolica di Milano)
The first two moments for the length of the period of the continued fraction expansion for sqrt{d}
This is a joint work with Loic Grenié and Giuseppe Molteni. Given a positive integer d which is not a square, denote by T(d) the length of the period of the continued fraction expansion for sqrt{d}. We prove upper bounds for the first and the second moment of T(d) by studying a different function g(d), originally introduced by Hickerson: we detect the asymptotic of the first moment of g(d) and an upper bound for the second moment. The results allow to improve the estimates for the size of the sets of integers d for which T(d) > alpha * sqrt{d}, with alpha a real parameter. We also report recent progress by Korolev on the asymptotic for the second moment of g(d).
2024.11.8. Andrea Ferraguti (Università di Torino)
Abelian dynamical Galois groups over function fields
Dynamical Galois groups are constructed by iterating a rational function over a field K and looking at the tower of preimages of a fixed point of P^1. A couple of years ago, Andrews and Petsche conjectured that when K is a number field and the function is a polynomial, such groups can only be abelian in trivial cases. This has only been proven to be true in a handful of cases, e.g. when K is the field of rationals. In this talk, we will consider the same question when K is a global function field. I will show how, combining a series of reduction steps that involve the construction of Bottcher coordinates in positive characteristic, it is possible to prove that if the rational function f has a superattracting cycle the dynamical Galois group of f with basepoint \alpha can be abelian only if the pair (f, \alpha) is defined over the constant field of K. This is joint work with P. Ingram and C. Pagano.
2024.11.15. Bianca Maria Gariboldi (Università degli Studi di Bergamo)
Irregularities of distribution and discrepancy theory: from Weyl up to nowadays
Starting from the Weyl criterion for uniformly distributed sets of points, we introduce the discrepancy theory, focusing on some classical results by Roth, Davenport, Cassels and Montgomery. We conclude the seminar with some new results on manifolds strictly related to the classical ones.