For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Q_p is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Q_p is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Q_p is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices. 

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy.