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...
Let A be an abelian variety over a number field E⊂ℂ and let v be a
place of good reduction lying over a prime p. For a prime ℓ≠p, a
result of Deligne implies that upon replacing E by a finite
extension, the Galois representation on the ℓ-adic Tate...
Fekete (1923) discovered the notion of transfinite diameter while
studying the possible configurations of Galois orbits of algebraic
integers in the complex plane. Based purely on the fact that the
discriminants of monic integer irreducible...
One of the fundamental challenges in number theory is to understand
the intricate way in which the additive and multiplicative
structures in the integers intertwine. We will explore a dynamical
approach to this topic. After introducing a new...
The classical Linnik problems are concerned with the
equidistribution of adelic torus orbits on the homogeneous spaces
attached to inner forms of GL2, as the discriminant of the torus
gets large. When specialized, these problems admit beautiful...
