#### PERIODS

##### Nisyros Island, Greece 6th-11th July 2023

## Joseph Ayoub: Periods: complex, functional, p-adic and almost all primes.

I will discuss several type of periods that appear naturally in algebraic geometry. In particular, I will recall how complex periods arise when considering an algebraic variety (or a motive) defined over the field of rational numbers. Then, I will explain a version of this construction in family yielding functional periods. I will then move to the p-adic situation and explain the construction of a pairing between de Rham cohomology and (rigid analytic) Suslin homology yielding interesting p-adic numbers that we call p-adic periods. (Special cases of these p-adic periods were studied by André and Ancona--Fratila.) Finally, I will discuss a very different type of periods living in the ring \Q_{pp}=\colim_N\prod_{p>N}\F_p, which are not, at least in an obvious way, of motivic origin. Along the way, we state several versions of the Grothendieck period conjecture, and explain what is known in each case.

## Raphael Beuzart-Plessis: Periods of automorphic forms and the relative Langlands program

This minicourse aims to introduce the audience to the fascinating subject of periods of automorphic forms and their relations to (special values of) L-functions. We will start with some examples where the automorphic periods under consideration can be interpreted as periods in the usual algebro-geometric sense (i.e. as integrals of closed forms on cycles) and then gently but surely move to a more representation-theoretic perspective. The goal would be to formulate some general predictions of Sakellaridis-Venkatesh that constitutes the core of what is now called "the relative Langlands program" and whose purpose is to understand the relations periods/L-functions in the automorphic context in a more systematic manner. We will provide, as we progress, background on automorphic forms and other related topics.

Jean-Benoît Bost: Infinite Dimensional Geometry of Numbers and Transcendence Proofs.

These lectures will present an approach to Diophantine geometry and transcendence proofs based on the recent development of some “infinite dimensional geometry of numbers,” where one studies infinite dimensional avatars of Euclidean lattices and their invariants, notably those defined in terms of the associated θ-series.

This formalism is systematically developed in the monograph [1], and applications to Diophantine geometry are given in [1] and [2].

In these lectures, I will focus on transcendence proofs, and present a proof of the theorem of Schneider-Lang, formally similar to the proof of classical algebraization results in complex analytic and formal geometry.

[1] J.-B. Bost. Theta invariants of euclidean lattices and infinite-dimensional hermitian vector over arithmetic curves. Progress in Mathematics, Vol. 334, Birkh Ìˆauser, 2020

[2] J.-B. Bost and F. Charles. Quasi-projective and Formal-analytic Arithmetic Surfaces. arXiv:2206.14242, 2022

Francis Brown: L-functions and periods

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After reviewing some classical examples of zeta and L-functions and their properties, I will discuss Deligne and Beilinson's conjectures which relate their values at integers to periods. Although the periods which arise as L-values in this way are very special, it turns out that they govern the structure of the ring of all periods.

I will then turn to Borel's theorem relating the values of zeta functions of number fields to algebraic K-theory, and the stable cohomology of the general linear group. This theorem is the cornerstone upon which crucially depends the existence, and key properties, of the known categories of mixed Tate motives. Borel's argument is highly transcendental, but I will explain, in the case of the integers, an algebraic-geometric construction which leads to a new version of Borel's argument. It leads to a definition of a motive associated to GL_n(Z) and provides much more powerful results on both the stable and unstable cohomology of GL_n(Z).

## Mattia Cavicchi: Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

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According to the Bloch-Beilinson conjectures, the vanishing of a motivic L-function at suitable integers should correspond to the existence of non-trivial extensions of motives. In this talk, I will describe work in progress with J. Bajpai, aiming to show a consequence of these predictions for a certain family of algebraic Hecke characters \phi of an imaginary quadratic number field. When the sign of the functional equation of L(\phi, s) is -1 - so that in particular, the L-function vanishes at the central point, and a period is expected to contribute to the transcendental part of the leading coefficient - we construct the extension of Hodge structures predicted by Bloch-Beilinson. For this, we employ the cohomology of Picard surfaces and the theory of Eisenstein cohomology due to G. Harder.

Tobias Kreutz: Motivic and arithmetic aspects of exceptional loci

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Given a family of smooth projective varieties defined over a number field, one often studies the Hodge locus (resp. the l-Galois exceptional locus), defined as the set of points in the base where exceptional Hodge (resp. l-adic Tate) classes occur in the cohomology of the fiber.

The Hodge and Tate conjectures state that Hodge and Tate classes should be classes of algebraic cycles, and therefore make the following predictions about these loci:

Both loci should in fact be equal, and moreover form a countable union of algebraic subvarieties of the base.

In this talk, I want to discuss some recent progress on these questions:

Firstly, we prove unconditionally that the l-Galois exceptional locus is indeed a countable union of algebraic subvarieties.

The corresponding result for the Hodge locus, obtained by Cattani-Deligne-Kaplan in 1995, is often viewed as important evidence in favor of the Hodge conjecture. In addition, under a mild condition on the generic Mumford-Tate group, we prove that the expected equality between the Hodge locus and the l-Galois exceptional locus holds true if one restricts to their positive dimensional parts.