The theme will be motivic/K-theoretic methods in number theory.
The speakers will be:
| 14:00-15:00 | Jesse Pajwani | Arithmetic information in a higher Euler characteristic For k a field the A1 Euler characteristic, constructed using motivic homotopy theory, furnishes a ring homomorphism K_0(Var_k) -> GW(k), which both refines the classical Euler characteristic of a CW complex and contains arithmetic information. Recent work by Nanavaty and Röndigs show that this ring homomorphism lifts to a morphism of spectra from the K theory spectrum of varieties, K(Var_k), to the endomorphisms of the motivic sphere spectrum over k. This in turn induces maps between higher homotopy groups of these spectra. In this talk, we study the induced morphism on the level of pi_1. We obtain an explicit homotopical description for this morphism, relate it to an invariant coming from Hermitian K theory, and give a few examples. This is joint work in progress with Ran Azouri, Stephen McKean and Anubhav Nanavaty. |
| 15:00-15:30 | coffee break (room 2L15) | |
| 15:30-16:30 | Thu-Hà Triệu | Mahler measure of polynomials and regulator The Mahler measure of polynomials was introduced by Mahler in 1962 as a tool to study transcendental number theory. In this talk, we discuss the relationship between Mahler measure and Beilinson's regulator. As an application, we show that, under some conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of special values of the elliptic curve L-function and the Bloch–Wigner dilogarithm. In certain four-variable cases, the Mahler measure can be written as a ℚ-linear combination of special values of the L-function of K3 surfaces and the Riemann zeta function. |
| 16:30-16:45 | short break | |
| 16:45-17:45 | Cecilia Busuioc | Modular Symbols with Values in Beilinson-Kato Distributions In this talk, we will describe the construction of a GL_n(ℚ) -invariant modular symbol with coefficients in a space of distributions that take values in Milnor K-groups of modular function fields. This is based on joint work with J. Park, O. Patashnick and G. Stevens. |
| 9:30-10:30 | Oliver Gregory | Codimension 2 algebraic cycles on supersingular varieties I will explore algebraic equivalence for codimension 2 algebraic cycles on supersingular varieties. I will explain that this conjecturally coincides with homological equivalence when the base field is the algebraic closure of a finite field and prove it in many cases. The takeaway should be yet more evidence that the geometry of supersingular varieties is surprisingly simple. |
| 10:30-10:50 | coffee break (room 2L15) | |
| 10:50-11:50 | Baptiste Morin | Zeta-values and the Beilinson fiber square We give a conjectural description of Zeta-values of arithmetic schemes in terms of two perfect complexes of abelian groups and a canonical trivialization. Then we state a (proven) archimedean analogue of this conjecture. In order to show compatibility with the classical Bloch-Kato conjecture, we study the Beilinson fiber square from a prismatic viewpoint, which provides precise integral information. This implies new cases of the Bloch-Kato conjecture. This is work in progress, joint with Matthias Flach and Achim Krause. |
| 11:50-14:00 | lunch break | |
| 14:00-15:00 | Ferdinand Wagner | Cohomology theories from refined THH Topological Hochschild homology (THH) has been very successful in constructing prismatic cohomology for p-adic formal schemes, but in analytic situations (e.g. for rigid-analytic varieties) it is less useful: THH can only extract rational information from rational inputs, and so it can never be used to recover, say, étale cohomology with torsion coefficients. In this talk, I'll explain a refinement of THH, due to Efimov--Scholze, which overcomes this issue. I'll also sketch how one can get cohomology theories (both for rigid-analytic varieties and varieties over Q) from this construction and how these are related to Scholze's Habiro cohomology. |
The organising committee currently consists of George Boxer (Imperial), Vladimir Dokchitser (UCL), Luis Garcia Martinez (UCL), Giada Grossi (Paris Nord), Marc Hindry (Jussieu), Stephen Lester (KCL), Matthew Morrow (Orsay), Beth Romano (KCL), Shu Sasaki (QMUL)
Here is the archive of past meetings.