Summer School: June 1721, 2019
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 BuinierKudlaYang 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. BeuzartPlessis (CNRS, Marseille)
Title : Introduction to the GanGrossPrasad and IchinoIkeda conjectures
Abstract : The aim of this minicourse will be to present the socalled global GanGrossPrasad and IchinoIkeda conjectures which, in broad terms, relate certain explicit integrals of automorphic forms (called 'automorphic periods') to special values of (automorphic) Lfunctions. 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 padic cohomology
Abstract : This minicourse will introduce the audience to some recent developments in integral padic Hodge theory, originating from the cohomology theory A\Omega introduced in the 2016 article "Integral padic Hodge theory". This provides a natural interpolation between de Rham, crystalline, and padic 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 qdeformation of de Rham cohomology via prisms (work in progress by BhattScholze). Depending on developments before the workshop, the course will most likely focus on some aspects of the relative theory, for example how modules with qconnection correspond to older notions in padic 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 pullback and
pushforward etc.
B. Schraen (CNRS, Paris)
Title : Companion forms for padic automorphic forms
Abstract : Let f be a eigenform on a definite unitary group. A companion
form of f is a padic 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 padic Hodge
theory described by Matthew Morrow in his course can be used to prove
classification theorems for pdivisible 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 ladic sheaves on varieties over curves
Abstract : Using the theory of singular supports and characteristic cycles for ladic sheaves, developed by Beilinson and Saito respectively, we studies the ramification bound of nearby cycles of ladic sheaves on schemes smooth over equal characteristic traits. We also discuss the ramification of ladic 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 padic Hodge theory
Abstract : We discuss connections between the Faltings height and
developments in padic 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 padic Simpson correspondence
Abstract : Deninger and Werner developed an analogue for padic curves of the classical correspondence of Narasimhan and Seshadri. Using parallel transport, they associated functorially to every vector bundle on a padic curve whose reduction is strongly semistable of degree 0 a padic representation of the étale fundamental group of the curve. We will explain how to use fundamental results in padic Hodge theory to understand their construction in the context of padic 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
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