Jun 17, 2019 to Jun 28, 2019

The program of the Summer School on Arithmetic Geometry that will take place in Carthage in the city of Tunis (Tunisia), from June 17 to 21, 2019 is now available at this address

It will be followed by a conference on the same topics from June 24-28, 2019. Further information is available at:

It will be followed by a conference on the same topics from June 24-28, 2019. Further information is available at:

Both the conference and the summer school will take place in the Tunisian Academy Beit al-Hikma located in a nice Palace from the middle of the 19th century, erected on an ancient archeological site at the foot of Carthage hill next to Antoninus Thermae and the Punic ports of Carthage, overlooking the sea.

Registration on the web is mandatory both for the Summer School and the Conference. We will close registrations when the maximum capacity of the conference hall is reached (a hundred participants).

Speakers don't need to register and are exempted from the registrationfees for both weeks.

Ahmed Abbes (CNRS, IHES), Christophe Breuil (CNRS, Orsay), Michael Harris (Columbia University), Ariane Mézard (Université Pierre et Marie Curie) and Takeshi Saito (University of Tokyo)

**Organizing Commitee:**

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

**Scientific Commitee:**

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

**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.

**Lectures** (1 hour each)

A.-C. Le Bras (Universität Bonn), H. Hu (MPIM, Bonn) J. Lin (Universität Duisburg-Essen),

L. Mocz (Princeton), D. Xu (Caltech), E. Yang (Peking University)

**Conference : June 24-28, 2019**

**Speakers**

A. Beilinson (University of Chicago), A. Caraiani (Imperial College), L. Fargues (CNRS, Paris),

J. Fresán (École Polytechnique), D. Gaitsgory (Harvard), W. T. Gan (National University of Singapore),

Q. Guignard (ENS & IHÉS), G. Henniart (Orsay), A. Ichino (Kyoto University), M. Kim (Oxford),

T. Koshikawa (RIMS, Kyoto University), M. Rapoport (Universität Bonn), K. Shimizu (UC Berkeley),

Y. Tian (Universität Bonn), T. Tsuji (University of Tokyo), M.-F. Vigneras (Sorbonne Université),

S. Zhang (Princeton), X. Zhu (Caltech)

Jun 17, 2019 to Jun 28, 2019

No participants