To any bounded family of \(\mathbb F_\ell\)-linear representations
of the etale fundamental of a curve \(X\) one can associate
families of abstract modular curves which, in this setting,
generalize the `usual' modular curves with level \(\ell\)...
Let \(k\) be an algebraically closed field and let \(c:C\rightarrow
X\times X\) be a correspondence. Let \(\ell \) be a prime
invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb
Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by...
I will begin with a brief introduction to the deformation theory
of Galois representations and its role in modularity lifting. This
will motivate the study of local deformation rings and more
specifically flat deformation rings. I will then discuss...
Let \chi be a primitive real character. We first establish a
relationship between the existence of the Landau-Siegel zero of
L(s,\chi) and the distribution of zeros of the Dirichlet L-function
L(s,\psi), with \psi belonging to a set \Psi of...
A continuous representation of a profinite group induces a
continuous pseudorepresentation, where a pseudorepresentation is
the data of the
characteristic polynomial coefficients. We discuss the geometry of
the resulting map from the moduli formal...
The theorem of the title is that if the L-function L(E,s) of an
elliptic curve E over the rationals vanishes to order r=0 or 1 at
s=1 then the rank of the group of rational rational points of E
equals r and the Tate-Shafarevich group of E is finite...
A few years ago Ichino-Ikeda formulated a quantitative version
of the Gross-Prasad conjecture, modeled after the classical work of
Waldspurger. This is a powerful local-to-global principle which is
very suitable for analytic and arithmetic...
The trace formula has been the most powerful and mainstream tool
in automorphic forms for proving instances of Langlands
functoriality, including character relations. Its generalization,
the relative trace formula, has also been used to prove...
Application of Plancherel's theorem to integral kernels
approximating compact period functionals yields estimates on
(global) automorphic Levi-Sobolev norms of the functionals. The
utility of this viewpoint can be illustrated in reconsideration
of...
Abstract:
Associated to an abelian variety A/K is a Galois representation
which describes the action of the absolute Galois group of K on the
torsion points of A. In this talk, we shall describe how large the
image of this representation can be (in...