Organisers: Giuseppe Ancona (Strasbourg), Amador Martin-Pizarro and Annette Huber (Freiburg), Philipp Habegger (Basel)
Date: March 28, 2025
Venue: Freiburg University, Mathematical Institute (Ernst-Zermelo-Straße 1); SR 404 (talks), Common Room 331 (coffee)
Program
10:15-11:00 Coffee reception
11:00-11:55 Umberto Zannier (Scuola Normale Pisa): On a problem of Polya and some of its evolutions
Around 1921 Polya posed a problem about algebraicity of the indefinite integral of an algebraic function, in terms of the integrality of the coefficients in a power-series expansion. This was solved much later by transcendence techniques. It also connects with a conjecture of Grothendieck on linear differential equations over polynomial rings, and with other number-theoretical issues. We shall survey on the topic, also illustrating a question related to a different possible approach to Polya's issue, and some answers given by Katz.
Lunch
14:00-14:55 Marco Artusa (Strasbourg): Topological duality for 1-motives over a p-adic field
Duality theorems are among the central statements in arithmetic geometry. For p-adic fields, the earliest example is Tate's duality for Galois cohomology of abelian varieties. The classical generalisation of this result to tori is not satisfying enough. This is due to certain shortcomings of Galois cohomology, such as the absence of a natural topology on cohomology groups. In this talk, we build a new topological cohomology theory for p-adic fields, thanks to the Weil group and Condensed Mathematics. We use it to extend Tate's theorem to 1-motives, improving a result by Harari and Szamuely. This new duality takes the form of a Pontryagin duality between locally compact abelian groups.
Coffee break
15:30-16:25 Alexei Skorobogatov (Imperial College London): Hasse principle without Schinzel
I will survey known cases of the Hasse principle conditional only on the finiteness of the Tate-Shafarevich group but not on Schinzel’s Hypothesis (H). A recent conditional result of this kind obtained jointly with Adam Morgan gives the Hasse principle for intersections of two quadrics of dimension 2 with trivial Brauer group and irreducible or completely split characteristic polynomial. The proof first establishes the Hasse principle for double covers which are Kummer surfaces attached to genus 2 curves.