We show that a positive proportion of cubic and quartic number fields are not monogenic, and not just for local reasons. We also show that a positive proportion of quartic number fields are not binary, i.e. they don't arise as the invariant order attached to a binary quartic form. To prove these results, we study rational points (and the lack thereof) in a family of elliptic curves. Joint work with Levent Alpoge and Manjul Bhargava.

I will present a general upper bound for the number of number fields of fixed degree and bounded discriminant. The method refines that of Ellenberg and Venkatesh, and improves upon their bound and that of Couveignes. This is joint work with Robert Lemke Oliver.

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruences. We give an algorithm for computing explicit instances of such congruences for eta-quotients, and we illustrate our method with a few examples.

Joint work with Nathan Ryan, Zachary Scherr and Stephanie Treneer.

In this talk we will discuss two invariants attached to a number field: the integral quadratic form associated to its trace paring and its shape. We prove that, as long as we restrict the ramification, these invariants are in fact strong enough to completely determine the isomorphism class of a field within the family of totally real Sn-number fields; as a byproduct of the method we are able to describe the automorphism group of the integral trace form for such fields. The chief ingredient in the proofs is a linear algebra gadget for quadratic forms we called “Casimir pairings”. They generalize the Casimir element from the theory of Lie algebras as well as the usual inner product of 1-forms in Riemannian geometry. For trace forms of number fields (and more generally of étale algebras), I will explain how the main usefulness of Casimir pairings lies in their functorial properties which make them compatible with both Galois-étale theory and base changes. This is joint work with Guillermo Mantilla-Soler.

In this talk we will recall the formulation of Serre's modular conjecture, and we will explain how some strong new modularity results can be used to give a simplified proof of it. This is a joint work with Luis Dieulefait.

A CM point in the moduli space of complex elliptic curves is a point representing an elliptic curve with complex multiplication. A classical result of William Duke (1988), complemented by Laurent Clozel and Emmanuel Ullmo (2004), states that CM points become uniformly distributed on the moduli space when we let the discriminant of the underlying ring of endomorphisms of these elliptic curves go to infinity. Since CM points are algebraic, it is possible to study p-adic analogues of this phenomenon. In this talk I will present a description of the p-adic asymptotic distribution of CM points in the moduli space of p-adic elliptic curves. This is joint work with Ricardo Menares (PUC, Chile) and Juan Rivera-Letelier (U. of Rochester, USA).

In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function \(d_k(f)\) over short intervals and over arithmetic progressions for the function field \(\mathbb{F}_q[T]\) to certain integrals over the ensemble of unitary matrices when \(q \rightarrow \infty\). We study two problems: the average over all the monic polynomials of fixed degree that yield a quadratic residue when viewed modulo a fixed monic irreducible polynomial \(P\), and the average over all the monic polynomials of fixed degree satisfying certain condition that is analogous to having an argument (in the sense of complex numbers) lying at certain specific sector of the unit circle. Both problems lead to integrals over the ensemble of symplectic matrices when \(q \rightarrow \infty\). We also consider analogous questions involving convolutions of the von Mangoldt function. This is joint work with Vivian Kuperberg.

We consider the family of normal octic fields with Galois group D4, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.