The work of Green, Tao and Ziegler can be used to prove existence
and approximation properties for rational solutions of the
Diophantine equations that describe representations of a product of
norm forms by a product of linear polynomials. One can...
The hyperbolic Ax Lindemann conjecture is a functional
transcendental statement which describes the closure of "algebraic
flows" on Shimura varieties. We will describe the proof of this
conjecture and its consequences for the André-Oort conjecture...
Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series
convergent in a neighborhood of zero in the complex plane. Every
scheme \(\cal X\) of finite type over \(R\) defines a complex
analytic space \({\cal X}^h\) over an open disc \(D\) of small...
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...
