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.


Organizers

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

CNRS IHES SMT CGCT
CGCT CGCT ICTP Compositio
Investissements d?Avenir Universite Paris-Saclay Tunisian Ministry of Higher Education and Scientific Research
MESRST

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.