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...
A classical result identifies holomorphic modular forms with
highest weight vectors of certain representations of SL2(ℝ). We
study locally analytic vectors of the (p-adically) completed
cohomology of modular curves and prove a p-adic analogue of...
We will give an explicit construction and description of a
supercuspidal local Langlands correspondence for any p-adic group G
that splits over a tame extension, provided p does not divide the
order of the Weyl group. This construction matches any...
Classically, heights are defined over number fields or
transcendence degree one function fields. This is so that the
Northcott property, which says that sets of points with bounded
height are finite, holds. Here, expanding on work of Moriwaki
and...
The Langlands program is a far-reaching collection of conjectures
that relate different areas of mathematics including number theory
and representation theory. A fundamental problem on the
representation theory side of the Langlands program is the...
A subset D of a finite cyclic group Z/mZ is called a "perfect
difference set" if every nonzero element of Z/mZ can be written
uniquely as the difference of two elements of D. If such a set
exists, then a simple counting argument shows that m=n2+n+1...
