Consider the function field F of a smooth curve over FqFq,
with q>2q>2.
L-functions of automorphic representations
of GL(2)GL(2) over FF are important objects for
studying the arithmetic properties of the field FF.
Unfortunately, they can be...
I will discuss some new results on the structure of Selmer
groups of finite Galois modules over global fields. Tate's
definition of the Cassels-Tate pairing can be extended to a pairing
on such Selmer groups with little adjustment, and many of
the...
Given a K3 surface XX over a number field KK, we
prove that the set of primes of KK where the geometric
Picard rank jumps is infinite, assuming that XX has
everywhere potentially good reduction. This result is formulated in
the general framework of...
Over the last decades, following works around the Pila-Wilkie
counting theorem in the context of o-minimality, there has been a
surge in interest around functional transcendence results, in part
due to their connection with special points...
Let λλ be the Liouville function
and P(x)P(x) any polynomial that is not a square. An open
problem formulated by Chowla and others asks to show that the
sequence λ(P(n))λ(P(n)) changes sign infinitely often. We
present a solution to this problem for...
Let GG be a reductive group over a number
field FF and HH a subgroup. Automorphic periods
study the integrals of cuspidal automorphic forms
on GG over H(F)∖H(AF)H(F)∖H(AF). They are often
related to special values of certain L functions. One of the...
The Generalized Ramanujan Conjecture (GRC) for GL(n) is a
central open problem in modern number theory. Its resolution is
known to yield several important applications. For instance, the
Ramanujan-Petersson conjecture for GL(2), proven by Deligne...
Let \o be an order in a totally real field, say F. Let K be an
odd-degree totally real field. Let S be a finite set of places of
K. We study S-integral K-points on integral models H_\o of Hilbert
modular varieties because not only do said varieties...
Following Bourgain, Gamburd, and Sarnak, we say that the Markoff
equation x2+y2+z2−3xyz=0 satisfies strong approximation at a prime
p if its integral points surject onto its Fp points. In 2016,
Bourgain, Gamburd, and Sarnak were able to establish...
