There will be talks in April and May 2026. The seminar will take place in Aula INdAM at 12.
2026.04.24. Valentijn Karemaker (University of Amsterdam)
Arithmetic invariants of supersingular abelian varieties
We will study the moduli space of abelian varieties in characteristic p and in particular its supersingular locus S_g. We will show when this locus is geometrically irreducible, thereby solving a “class number one problem” or “Gauss problem” for the number of irreducible components; and when a polarised abelian variety is determined by its p-divisible group, solving a Gauss problem for central leaves, which are the loci consisting of points whose associated p-divisible groups are geometrically isomorphic. Furthermore, we will discuss Oort's conjecture, which states that all generic points of S_g have automorphism group {+/- 1}.
This is based on joint works with Ibukiyama and Yu.
2026.05.08. Diana Mocanu (Max Planck Institute of Mathematics)
Local points on twists of X(p)
Let E be a rational elliptic curve and p an odd prime. The modular curve X_E^{-}(p) parametrizes elliptic curves with p-torsion modules anti-symplectically isomorphic to E[p]. In this talk, I present my recent work with Nuno Freitas on a complete classification for when these curves admit points everywhere locally.
We will see two applications of this result. Firstly, I will show how to construct counterexamples to Hasse’s principle of the shape X_E^{-}(p) for fixed E and infinitely many primes p. Secondly, I will present an application of the modular method together with our results to prove certain generalized Fermat equations have no non-trivial coprime solutions.
2026.05.15. Anna Pippich (University of Konstanz)
Special Values of Green’s Functions on Hyperbolic 3-Space
In 1986, Gross and Zagier formulated an algebraicity conjecture concerning special values of higher Green’s functions on modular curves, revealing a deep analogy with the behavior of the modular j-invariant in the theory of complex multiplication. An averaged version of this conjecture was later proved by Gross, Kohnen, and Zagier, and the full conjecture has recently been established by Bruinier, Li, and Yang.
In this talk, I will present new results on an analogous problem in a three-dimensional setting, namely the special values of Green’s functions on hyperbolic 3-space. We show that certain averages of these values can be expressed in terms of logarithms of primes and logarithms of units in real quadratic fields. We also discuss twisted averages that give rise to algebraic numbers. This is joint work with Özlem Imamoglu, Sebastian Herrero, and Markus Schwagenscheidt.
2026.05.22. Daniele Bartoli (Università di Perugia)
Exceptional scattered polynomials
Scattered polynomials represent a specialized class of linearized polynomials over finite fields that are fundamental to the study of maximum scattered linear sets and the construction of maximum rank distance (MRD) codes. A q-polynomial f(X) is categorized as a scattered polynomial of index t if the associated vector subspace U = { (x^{q^t}, f(x)) : x ∈ F_{q^n} } defines a maximum scattered linear set in PG(1, q^n). This presentation explores the classification of exceptional scattered polynomials, which are characterized by their ability to maintain the maximum scattered property over infinitely many field extensions.
While the classification of these rare polynomials can be approached through group-theoretical results involving the Galois groups of linearized polynomials, this talk focuses on a geometric and analytical framework. This methodology involves mapping polynomial properties to the study of singular points on associated algebraic curves. By employing local quadratic transformations, branch investigation, and the Hasse-Weil Theorem, we establish precise bounds on intersection multiplicities to identify absolutely irreducible components. These geometric techniques provide the necessary conditions to definitively classify known families and clarify the asymptotic behavior of exceptional scattered polynomials.
2026.06.05. Giacomo Micheli (University of South Florida)
On Some Applications of the theory of Drinfeld Modules
In this talk we apply the theory of Drinfeld modules to construct rank metric codes, which are important information-theoretical objects. In particular, we focus on constructions of MRD codes and optimal rank metric codes with rank locality.
Past Events