We compare absolute and relative Gromov-Witten invariants with the
basic contact vector for very positive divisors. For such divisors,
one might expect that these invariants are the same up to a natural
multiple. We show that this is indeed the case...
Based on a new decoupling inequality for curves in $\mathbb R^d$,
we obtain the essentially optimal form of Vinogradov's mean value
theorem in all dimensions (the case $d = 3$ is due to T. Wooley).
Various consequences will be mentioned and we will...
Let $G$ be a reductive group over a global field of positive
characteristic. In a major breakthrough, Vincent Lafforgue has
recently shown how to assign a Langlands parameter to a cuspidal
automorphic representation of $G$. The parameter is a...
For a representation of the absolute Galois group of the rationals
over a finite field of characteristic $p$, we would like to know if
there exists a lift to characteristic zero with nice properties. In
particular, we would like it to be geometric...
I will speak about results contained in my article "$G$-torseurs en
théorie de Hodge $p$-adique" linked to local class field theory. I
will in particular explain the computation of the Brauer group of
the curve and why its fundamental class is the...
Given the $p$-adic Galois representation associated to a regular
algebraic polarized cuspidal automorphic representation, one
naturally obtains a pure weight zero representation called its
adjoint representation. Because it has weight zero, a...
Venkatesh has recently proposed a fascinating conjecture relating
motivic cohomology with automorphic forms and the cohomology of
arithmetic groups. I'll describe this conjecture, and discuss its
connections with the local geometry of eigenvarieties...
Starting from the Poisson summation formula, I discuss spectral
summation formulae on GL(2) and GL(3) and present a variety of
applications to automorphic forms, analytic number theory, and
arithmetic.
The formal degree conjecture relates the formal degree of an
irreducible square-integrable representation of a reductive group
over a local field to the special value of the adjoint gamma-factor
of its L-parameter. We prove the formal degree...
The endoscopy theory provides a large class of examples of
Langlands functoriality, and it also plays an important role in the
classification of automorphic forms. The central part of this
theory are some conjectural identities of Harish-Chandra...
