Speaker: Aleksander Horawa (University of Oxford, United Kingdom)
Title: Siegel modular forms and higher algebraic cycles
Abstract: The motivic action conjecture of Akshay Venkatesh and collaborators predicts a subtle connection between the cohomology of locally symmetric spaces and motivic cohomology groups (a.k.a. higher Chow groups). The case of Siegel modular threefolds is particularly interesting because Siegel modular forms of weight 2 are expected to correspond to rational abelian surfaces just as modular forms of weight 2 correspond to rational elliptic curves. In joint work with Kartik Prasanna we gave a statement of the conjecture in this case and proved it in special cases. In ongoing work with Lambert A’Campo and Hohto Bekki we construct the relevant motivic cohomology classes for Jacobians of genus 2 curves and compute their regulators.
Speaker: Pengju Guan (Vrije Universiteit Amsterdam)
Title:
Computing Hasse-Weil zeta functions via
Abstract: The number of rational points of an algebraic variety defined over a finite field is very interesting information, which is encoded in the Hasse-Weil zeta function. In this talk, I will try to explain how to use differential 1-forms of the second kind, together with a lift of Frobenius, to compute the zeta function of some algebraic curves. This is joint work in progress with Amnon Besser, Rob de Jeu, Muxi Li and Nicolas Mascot.
Speaker: Rob de Jeu (Vrije Universiteit Amsterdam)
Title: Conjectures (and results) on p-adic Artin L-functions
Abstract: After an introduction to and review of such p-adic L-functions, we discuss conjectures and results (partly computational) on their zeroes. This includes non-vanishing at non-zero integers, which is motivated by applications, and itself generalises the Leopoldt conjecture (for totally real number fields). We also discuss some statistical evidence for the behaviour of the constant term of the underlying Iwasawa series in certain cases. This is joint work with X.-F. Roblot.
Speaker: Pol van Hoften (Vrije Universiteit Amsterdam)
Title: p-adic Fourier theory for p-divisible rigid analytic groups
Abstract: I aim to give a leisurely introduction to p-adic Fourier theory such as the Mahler transform and the Amice transform. I will explain how these transforms can be formulated geometrically in terms of the formal (and analytic) multiplicative p-divisible group. I will then discuss generalizations of these transforms to general p-divisible rigid analytic groups. This is work in progress with Andrew Graham and Sean Howe.
Speaker: Sarah Zerbes (ETH Zurich, Switzerland)
Title: Iwasawa theory for the symmetric square of an elliptic curve
Abstract: The arithmetic of the adjoint, or symmetric square, of an elliptic curve over Q (or, more generally, of a modular form) is a particularly interesting case from the viewpoint of Iwasawa theory, not least because of its close connection with modularity-lifting problems and hence with Fermat's last theorem. In this talk I will describe joint work with David Loeffler in which we prove the cyclotomic Iwasawa main conjecture in this setting, using Euler systems for Hilbert modular surfaces.
Speaker: David Loeffler (UniDistance Suisse, Switzerland)
Title: Poles of p-adic Asai L-functions and base change
Abstract: If F is a Hilbert modular form over a quadratic field, then one can characterise whether or not F is a base-change lift from Q via L-functions: roughly, the Asai L-function of F has a pole if and only if F is a base-change form. This is a prototype for many more general results characterising the images of functorial lifts of automorphic forms via poles of their L-series.
In a recent work with Grossi and Zerbes, we defined a p-adic analogue of the Asai L-function; but, surprisingly, this p-adic L-function is always an analytic function - it never has poles, whether or not F is a base-change. I will explain a solution to this puzzle, by defining an "improved" p-adic L-function associated to a Hida family of Hilbert modular forms, whose interpolation property involves fewer Euler factors than the usual one; and showing that this new object has a pole if and only if F is a base-change of the expected type, giving a p-adic analytic characterisation of the image of the base-change lift.
This is joint work with Sarah Zerbes.
Speaker: Remy van Dobben de Bruyn (Universiteit Utrecht, The Netherlands)
Title: Constructible sheaves on toric varieties
Abstract: Given a reasonable topological space or algebraic variety, its covering spaces are classified by the fundamental group, via the monodromy correspondence. Recently, this was upgraded to an exodromy correspondence, classifying constructible sheaves as representations of a 'stratified fundamental category'. I will show how to prove this for toric varieties over an arbitrary field, including an explicit determination of the fundamental category. Over the complex numbers, this is a simplification of a result of Braden and Lunts from 2006.
Speaker: Greg Pearlstein (University of Pisa, Italy)
Title: Infinitesimal Torelli and rigidity results for a remarkable class of elliptic surfaces
Abstract: I will discuss joint work with Chris Peters which extends rigidity results of Arakalov, Faltings and Peters to period maps arising from families of complex algebraic varieties which are non-necessarily proper or smooth. Inspired by recent work with P. Gallardo, L. Schaffler, Z. Zhang, I will discuss two classes of elliptic surfaces which can be presented as hypersurfaces in weighted projective spaces which have a unique canonical curve. In each case, we will show that infinitesimal Torelli fails for H2 of the compact surface, but is restored when one considers the period map for the complement of the canonical curve.
Speaker: Vasilii Bolbachan (Skolkovo Institute of science and technology, Russia)
Title: Goncharov's conjecture and higher Chow groups
Abstract: Goncharov's conjecture states that the motivic cohomology of a field is isomorphic to the cohomology of the polylogarithmic complex. The talk will be devoted to the proof of this conjecture in the case when the degree is one less than the motivic weight. The main ingredient used in the proof is some new version of higher Chow groups. I want to explain the main properties of this complex and discuss some open questions.
Speaker: Lambert A'Campo (MPIM Bonn, Germany)
Title: Regulators and the Motivic Cohomology of Fermat Hypersurfaces
Abstract: Fermat hypersurfaces give rise to an interesting set of examples of motives. Beilinson's conjecture relates special values of the corresponding L functions to regulators on motivic cohomology groups. I will talk about making this relation as explicit as possible. As a special case we obtain an integral formula for the Apery constant which we have verified numerically up to 15 digits. This is joint work in progress with Aleksander Horawa.
Speaker: David Lilienfeldt (Universiteit Leiden, The Netherlands)
Title: Experiments with Ceresa classes of cyclic Fermat quotients.
Abstract: Let C be a curve of genus g embedded in its Jacobian J. The Ceresa cycle C-[-1]*C is a homologically trivial algebraic cycle of codimension g-1 on J. When C is hyperelliptic, this cycle is trivial modulo algebraic equivalence, whereas for a general curve C it is non-trivial by work of Ceresa. The first example of a non-hyperelliptic curve with torsion Ceresa class modulo algebraic equivalence was found by Beauville and Schoen. We give two new examples of curves with such behavior. All three examples (including the one of Beauville and Schoen) are cyclic quotients of Fermat curves. We compute the central order of vanishing of the L-functions of the relevant motives and give evidence in one case for the conjecture of Beilinson-Bloch. This is joint work with Ari Shnidman.