Summer School: June 17-21, 2019

Carthage Beit al Hikma Ports puniques Sidi Bou Said

Titles and abstracts of the courses (3 hours each)

F. Andreatta (Università Statale di Milano)
Title : Faltings heights of abelian varieties with complex multiplication
Abstract : After introducing Shimura varieties of orthogonal type, their Heegner divisors and the so called CM (complex Multiplication) points, I will review a conjecture of Buinier-Kudla-Yang which provides explicit formulas for their arithmetic intersection. I will show that they imply an averaged version of a conjecture of Colmez on the height of CM abelian varieties.

R. Beuzart-Plessis (CNRS, Marseille)
Title : Introduction to the Gan-Gross-Prasad and Ichino-Ikeda conjectures
Abstract : The aim of this mini-course will be to present the so-called global Gan-Gross-Prasad and Ichino-Ikeda conjectures which, in broad terms, relate certain explicit integrals of automorphic forms (called 'automorphic periods') to special values of (automorphic) L-functions. Once properly introduced, we will survey recent progress on these conjectures, most notably by W. Zhang, as well as remaining open problems.

M. Morrow (CNRS, Paris)
Title : Recent developments in integral p-adic cohomology
Abstract : This mini-course will introduce the audience to some recent developments in integral p-adic Hodge theory, originating from the cohomology theory A\Omega introduced in the 2016 article "Integral p-adic Hodge theory". This provides a natural interpolation between de Rham, crystalline, and p-adic étale cohomology. It may be constructed in several different fashions: either as a modification of Galois/étale cohomology, or via topological cyclic homology, or most recently (and most generally) in its guise as a canonical q-deformation of de Rham cohomology via prisms (work in progress by Bhatt-Scholze). Depending on developments before the workshop, the course will most likely focus on some aspects of the relative theory, for example how modules with q-connection correspond to older notions in p-adic Hodge theory such as Faltings? generalised representations and relative Fontaine modules (joint with T. Tsuji).

T. Saito (University of Tokyo)
Title : Characteristic cycle of a constructible sheaf
Abstract : We discuss basic ingredients in the definition of the singular support and the characteristic cycle of a constructible sheaf on a smooth scheme over a field of positive characteristic. We also discuss their main properties including the index formula, functoriality for pull-back and push-forward etc.

B. Schraen (CNRS, Paris)
Title : Companion forms for p-adic automorphic forms
Abstract : Let f be a eigenform on a definite unitary group. A companion form of f is a p-adic eigenform which has same prime to p Hecke eigenvalues than f. Companion forms can be non classical and can be of weight different from the weight of f. In the lectures, I will explain how companion forms of f can be predicted by the Galois representation associated to f when f has a level prime to p and give a sketch of the proof in some particular cases, following a work in collaboration with Breuil and Hellmann. The proof is linked to a local analogue for trianguline Galois representations. Here is a provisional program for the three hours :
Lecture 1 : statement of the result on companion forms.
Lecture 2 : trianguline local representations and trianguline variety.
Lecture 3 : a sketch of proof of the global result.

Titles and abstracts of the lectures (1 hour each)

A.-C. Le Bras (CNRS & Université Paris 13)
Title : Prismatic Dieudonné theory
Abstract : I will explain how the recent developments in p-adic Hodge theory described by Matthew Morrow in his course can be used to prove classification theorems for p-divisible groups, both in characteristic p and in mixed characteristic. This is based on joint work with Johannes Anschütz.

H. Hu (Nanjing University)
Title : Ramification of l-adic sheaves on varieties over curves
Abstract : Using the theory of singular supports and characteristic cycles for l-adic sheaves, developed by Beilinson and Saito respectively, we studies the ramification bound of nearby cycles of l-adic sheaves on schemes smooth over equal characteristic traits. We also discuss the ramification of l-adic sheaves along a vertical divisor of relative curves and abelian schemes over curves following an idea of T. Saito. This is a joint work with J.-B. Teyssier.

L. Mocz (Universität Bonn)
Title : Heights and p-adic Hodge theory
Abstract : We discuss connections between the Faltings height and developments in p-adic Hodge theory. One of the fundamental results stemming from this connection is the proof of a new CM Northcott property for the Faltings height.

D. Xu (Caltech)
Title : Parallel transport and p-adic Simpson correspondence
Abstract : Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the étale fundamental group of the curve. We will explain how to use fundamental results in p-adic Hodge theory to understand their construction in the context of p-adic Simpson correspondence developed by Faltings.

E. Yang (Peking University)
Title : Twist formula of epsilon factors of constructible étale sheaves
Abstract : Using Beilinson and T. Saito's theory, we prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. This formula is a modified version of a conjecture by Kato and T. Saito. We also propose a relative version of the twist formula and discuss some applications. This is a joint work with Naoya Umezaki and Yigeng Zhao.


A. Abbes (CNRS & IHÉS), C. Breuil (CNRS, Orsay), M. Harris (Columbia University),
A. Mézard (Sorbonne Université), T. Saito (University of Tokyo)

Organized in partnership with

Investissements d?Avenir Universite Paris-Saclay Tunisian Ministry of Higher Education and Scientific Research

Registration on the web is mandatory both for the Summer School (17-21 June 2019) and the Conference (24-28 June 2019).
Please fill the registration form here.