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...

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Joint IAS/PU Number Theory

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...

### Strong approximation for the Markoff equation via nonabelian level structures on elliptic curves

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(?). 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...