Exceptional loci in Lang-Vojta Conjectures

• Amos Turchet (Università Roma Tre)

Room: Saletta Riunioni The distribution of integral points over number fields in algebraic varieties is conjecturally described by far reaching conjectures of Lang and Vojta. In the case of surfaces of log general type, it predicts the existence of a proper closed subset, the “exceptional set”, that contains all but finitely many integral points, on every finite extension of the ground field. In this talk we will discuss the exceptional set for the complement in the projective plane of a curve with at least three components. We will completely describe the “algebraic" exceptional set, how it relates to so-called hyperbitangent curves, the implication for (Brody)-hyperbolicity and, time permitting, generalization to arbitrary surfaces. This is joint work with Lucia Caporaso.

Minoration of the Weil height over p-adic Lie extensions

• Lea Terracini (Università di Torino)

Room: Saletta Riunioni I will discuss recent progress on the Bogomolov property for algebraic extensions cut out by p-adic Galois representations, with no modularity assumptions. Under a big local image condition on inertia at p, Sen’s comparison between ramification and Lie filtrations allows to produce a uniform p-adic metric inequality, which in turn leads to a global lower bound for the Weil height. If time allows, I will illustrate the criterion through both modular and non-modular examples. This is joint work with Andrea Conti.

Galois groups giving counterexamples to the Hasse principle for divisibility in elliptic curves

• Laura Paladino (Università della Calabria)

Room: Saletta Riunioni Let $p$ be a prime number and let $n$ be a positive integer. Let $\mathcal E$ be an elliptic curve defined over $k$. The validity or the failing of the Hasse principle for divisibility in ${\mathcal E}/k$ depends on the Galois group $\textrm{Gal}(k({\mathcal E}[p^n])/k)$. For which kind of these groups does the principle hold? For which of them can we find a counterexample? The answers to these questions were known for $n = 1, 2$, but for $n \geq 3$ they were still open. We now present best possible answers for every $n$, by showing conditions on the generators of $\textrm{Gal}(k({\mathcal E}[p^n])/k)$, which are sufficient and necessary for the validity of the local-global principle. This is a joint work with Jessica Alessandr\`i (Max Planck Institute Bonn).

Drinfeld quasi-modular forms of higher level

• Maria Valentino (Università della Calabria)

Room: Saletta Riunioni Quasi-modular forms, first introduced by Kaneko and Zagier in the 90's, arise as a natural extension of classical modular forms and exhibit remarkable closure properties under differentiation. In this talk, we will explore their function field counterpart. Specifically, we will focus on the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups providing two different representations: one as polynomials in the false Eisenstein series and the other, whenever possible, as sums of hyperderivatives of Drinfeld modular forms. Finally, we will present the double-slash operator, which allows us to provide a well-posed definition for Hecke operators on Drinfeld quasi-modular forms, and a characterization of Hecke eigenforms.

Singular intersections of curves and subgroups in tori and in abelian varieties

• Laura Capuano (Università Roma Tre)

Room: Saletta Riunioni It can be proved that, if X is an irreducible curve in a torus defined over a number field, then the set of all the points of X lying in a proper algebraic group is always infinite, even under the hypothesis that X is not contained in a proper algebraic group. In a joint work with F. Ballini and N. Ottolini we prove that, if one looks at the multiplicities of intersections, the set of points where this intersection is singular is a finite set. This statement, which fits in the general framework of problems of Unlikely Intersections, generalizes a previous result of Marché and Maurin for curves in a torus of dimension 2.

On the disk of convergence of algebraic power series

• Francesco Veneziano (Università di Genova)

Room: Saletta Riunioni In a recent work with U. Zannier we considered the disk of convergence of a power series $s(x)$ representing an algebraic function of x and specifically the relation between this disk and the branch points of the $x$-coordinate on the corresponding algebraic curve. We focus especially on the $p$-adic case, answering some questions of basic nature, seemingly absent from the existing literature. To illustrate the issues, recall that in the complex case the open disk of convergence cannot contain all the branch points of $x$, except for the trivial case of a rational function. In the p-adic case, we show that the analogous assertion is not true in complete generality; but we also confirm it in a number of cases, for instance under the assumption that p is not smaller than the degree of $s(x)$ over the field of rational functions of $x$.