Schedule

Antonio Cauchi, Universitat Politècnica de Catalunya

Title: Algebraic cycles for the Siegel sixfold and the exceptional theta lift from G2

Abstract: In this talk, we will report some progress towards the Beilinson conjectures for Shimura varieties associated to the symplectic group GSp(6).
We will describe a cohomological formula for the residue at s=1 of the degree 8 spin L-function. We will then discuss an important family of cuspidal automorphic representations for PGSp(6) for which the residue is non-zero and relate this to the existence of an algebraic cycle coming from a Hilbert modular subvariety. This relation partially answers a question of Gross and Savin on motives with Galois group of type G2.
This is joint work with Francesco Lemma and Joaquin Rodrigues Jacinto.

Yukako Kezuka, Max-Planck-Institut für Mathematik

Title: On the non-triviality of the 2-part of the Tate-Shafarevich group

Abstract: The conjecture of Birch and Swinnerton-Dyer concerns a deep connection between the arithmetic of elliptic curves and the behaviour of their associated complex L-functions at s=1.
The conjecture was formulated in the early 60's, and much of it remains mysterious today.
Indeed, the exact Birch-Swinnerton-Dyer formula remains unknown even for the classical family of elliptic curves E of the form x^3+y^3=N, where N is a positive integer.

In this talk, I will study the "p-part" of the conjecture for these curves at small primes p.
These cases are often eschewed, but they seem to make up a most significant part of the full conjecture.
First, I will study the 3-adic valuation of the algebraic part of their central L-values, and use it to show that the "analytic" order of the Tate-Shafarevich group of E is a perfect square for some N.
In the second part of the talk, I will explain how we can obtain the 3-part of the Birch-Swinnerton-Dyer conjecture in certain special cases of N where the rank of E is known to be equal to 0 or 1. For the 2-part of the conjecture, I will explain a relation between the ideal class group of a corresponding cubic field extension and the 2-Selmer group of E.
This can be used to study non-triviality of the 2-part of the Tate-Shafarevich group of E, even when E has rank 1. The second part of this talk is joint work with Yongxiong Li.

Alex Panetta, Université Paris Diderot

Title: Higher regulators and special values of the degree-eight L-function of GSp(4)xGL(2)

Abstract: In order to prove Beilinson conjectures, we link the image of an element through the Beilinson regulator in the Deligne cohomology of the product of a Siegel variety and a modular curve respectively, to the special value in 1 of the degree-eight L-function of GSp(4)xGL(2) associated to a product of automorphic generic admissible cuspidal representations of GSp(4) and GL(2) respectively, in the case where this function is entire.
In this talk, we will explain how we can link these different objects using a linear form defined on the Deligne cohomology.

Andrew Graham, University College London

Title: Euler systems and p-adic L-functions for conjugate self-dual representations

Abstract: In this talk, I will describe joint work with S.W.A. Shah on the construction of a split anticyclotomic Euler system for a large class of conjugate self-dual automorphic representations admitting a Shalika model.
This Euler system arises from special cycles on unitary Shimura varieties and the proof of the norm relations amounts to a computation in local representation theory.
I will also describe the expected relation with p-adic L-functions (using the machinery of higher Hida theory) and (expected) applications to the Bloch--Kato conjecture.

Óscar Rivero, University of Warwick

Title: Eisenstein congruences and Euler systems

Abstract: Let f be a cuspidal eigenform of weight two, and let p be a prime at which f is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a relationship with the first derivative of the L-series of f at the near central point s=0. I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the Beilinson class with the arithmetic of circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson--Kato elements and, most crucially, the work of Fukaya--Kato.

Xiaoyu Zhang, Universität Duisburg-Essen

Title: p-part Bloch-Kato conjecture for Siegel modular forms of genus 2

Abstract: The Bloch-Kato conjecture relates the algebraic part of special L-values to the Selmer groups of the same motive.
In this talk, we study the p-part of this conjecture for a Siegel modular form of genus 2 and show, under mild conditions on the associated Galois representation, that the special value of the standard L-function divided by an automorphic period is equal to the characteristic ideal of the corresponding Selmer group, up to p-units.
The proof relies on some non-vanishing results of mod p theta lifts from the orthogonal group to the symplectic group.

Quentin Gazda, Université Claude Bernard Lyon 1

Title: First Beilinson’s conjecture in function fields arithmetic

Abstract: In the mid 80’s, Beilinson formulated deep conjectures relating special values of L-functions to pieces of K-theory, superseding at once the BSD conjecture and Deligne’s conjecture. Beilinson’s conjectures are fully expressed in the framework of mixed motives, which remains hypothetical.

This talk will be devoted to portray the analogous picture in the function fields setting, using so-called Goss L-values instead of classical L-values, and mixed (uniformizable) Anderson A-motives instead of Grothendieck's mixed motives. After a recall of the classical conjectures, we shall discuss and define the analogue of motivic cohomology and regulators for function fields, and express the counterpart of Beilinson’s conjectures.

Huy Hung Le, Université de Caen Normandie

Title: On identities for zeta values in Tate algebras.

Abstract: Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic.

In this talk, we will present some related conjectures proposed by Pellarin.
Then, we will study the Bernoulli-type polynomials attached to these zeta values.
By a combinatorial method, we can also provide some explicit formulas.
We will demonstrate how to use these results to prove a conjecture of Pellarin on identities for zeta values in Tate algebras.

Nils Matthes, University of Oxford

Title: A new approach to multiple elliptic polylogarithms

Abstract: Multiple polylogarithms may be viewed as the monodromy of a certain "universal" unipotent differential equation on the projective line minus three points. This observation lies at the heart of their relation to mixed Tate motives, a point of view which brings powerful new tools to bear on the study of these functions and its special values.

The goal of this talk is to describe an analogous picture for a once-punctured elliptic curve E'. In particular, we obtain a new description of the unipotent de Rham fundamental group of E', generalizing and improving on previous works of Levin-Racinet, Brown-Levin, Enriquez-Etingof, and others. Joint work in progress with Tiago J. Fonseca (Oxford).

Federico Zerbini, IPhT CEA-Saclay

Title: New modular forms from string theory, and single-valued periods

Abstract: I will introduce a class of modular forms, called modular graph functions, which originate from the computation of Feynman integrals in string theory.
Modular graph functions generalise real analytic Eisenstein series, their expansion coefficients are multiple zeta values, and they are conjecturally related to the theory of single-valued periods, which I will briefly review.
In particular, the expansion coefficients are conjectured to belong to a small subalgebra of the multiple zeta values whose elements are single-valued periods.

I will present a proof of this conjecture for the simplest kind of Feynman integrals, obtained in collaboration with Don Zagier.
I will also mention how modular graph functions are expected to be related to iterated extensions of pure motives of modular forms, and how one can attach L-functions to them.

Adam Keilthy, Max-Planck-Institut für Mathematik

Title: Block graded relations among motivic multiple zeta values

Abstract: Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables.

While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

Mahya Mehrabdollahei, Sorbonne Université

Title: Mahler measure of a family of exact polynomials

Abstract: I will present results around the Mahler measure of a family of 2-variate exact polynomials. The closed formula for the Mahler measure of two-variable exact polynomials gives an expression of each of these Mahler measures as a finite sum of the values of Dilogarithm at certain roots of unity. This allows:

• to compute their values with any precision;

• to use the techniques of Riemann sums to compute the limit of this sequence of Mahler measures and an asymptotic expansion, with a link with Boyd-Lawton theorem;

• to relate, for small values of d, these Mahler measures to values of special values of L-function, with a link with works of Boyd-Rodriguez Villegas and others.

Eugenia Rosu, Universität Regensburg

Title: Twists of elliptic curves with CM

Abstract: We consider certain families of sextic twists of the elliptic curve y^2=x^3+1 that are not defined over ℚ, but over ℚ[√-3].

We compute a formula that relates the value of the L-function L(E_D, 1) to the square of a trace of a modular function at a CM point.
Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we should recover the order of the Tate-Shafarevich group, and under certain conditions we show that the value is indeed a square.

Matteo Tamiozzo, Imperial College

Title: Torsion in the cohomology of Hilbert modular varieties (with a view towards Iwasawa theory)

Abstract: Yiwen Zhou has given a new construction of Kato's zeta element in local Iwasawa cohomology based on local-global compatibility between completed cohomology of modular curves and the p-adic Langlands correspondence. After recalling this construction, we will discuss the proof of a vanishing theorem for cohomology of Hilbert modular varieties which plays a key role in extending the above local-global compatibility result. This is joint work in progress with Ana Caraiani.