Joint IAS/PU Number Theory

We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical “quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is...

I will give an overview of recent progress in random multiplicative functions (random models for multiplicative functions) and their connection to the study of the Fyodorov-Hiary-Keating conjecture and Polya's question on nonnegative character sums...

Central predictions of arithmetic quantum chaos such as the Quantum Unique Ergodicity conjecture and the sup-norm problem ask about the mass distribution of automorphic forms, most classically in terms of their weight or Laplace eigenvalue (for...

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit...

How many algebraic integers of bounded height have a minimal polynomial with a given Galois group? One approach to this problem is via Malle's conjecture. In this talk we will discuss an alternative approach using a construction with the Galois...

A construction of Thurston assigns a hyperbolic 3-manifold to any polyhedron; a natural question is: which such are arithmetic? We report on ongoing work aiming to answer this question.

I will discuss recent results for algebraicity of critical values of Spin L-functions for GSp_6. I will also discuss ongoing work toward the construction of p-adic L-functions interpolating these values. I will explain how this work fits into the...

We show that for every quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. By work of Shlapentokh, this implies that Hilbert's tenth problem over the ring of...

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several...

We prove an effective equidistribution theorem for semisimple

closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. As an...