Georgia Institute of Technology College of Sciences

Given a modular form $f$, one can construct a measure $\mu_f$ on the modular surface $SL(2,\mathbb{Z})\backslash\mathbb{H}$. The celebrated mass equidistribution theorem of Holowinsky and Soundararajan states that as $k\rightarrow\infty$, the measure $\mu_f$ approaches the uniform measure on the surface. Given a maximal order in a quaternion algebra which is non-split over $\mathbb{Q}$, a maximal order leads to a cocompact subgroup of $R^1\subseteq SL(2,\mathbb{Z})$ where the quotient $R^1\backslash\mathbb{H}$ is a Shimura curve. Given a Hecke form $f$ on this Shimura curve, one can construct the analogous measure $\mu_f$, and ask about the limit as $k\rightarrow\infty$. Recent work of Nelson relates this equidistribution problem for the cocompact case to obtaining bounds on sums of Hecke eigenvalues summed over quadratic progressions. In this talk, I will describe this problem in both the cocompact and non-cocompact case while highlighting how differences in algebras lead to differences in geometry. I will then state progress that I have made on bounds that correspond to square root cancellation on average for sums of Hecke eigenvalues summed over quadratic progressions when averaged over a basis of Hecke forms. 

 For a positive integer , define  to be the smallest number such that the additive energy of any subset and any  is at most . In this talk, I will survey recent results on bounds for , explore the connections with (variants of) the Hausdorff-Young inequality in analysis and with the Balog-Szemeredi-Gowers theorem in additive combinatorics, and then discuss new results on the asymptotic behavior of  as .

Given a random polynomial f of degree n with integer coefficients each drawn uniformly and independently from an interval [-H, H], what is the probability that the Galois group of the roots of f is NOT the full symmetric group Sₙ? In 1936, van der Waerden conjectured that the answer should be of order 1/H, with the dominant contribution coming from f with a rational root. This conjecture was finally resolved by Bhargava in 2023. In this project (joint w/ Theresa Anderson), we ask the same question for reciprocal (a.k.a. palindromic) polynomials, which arise for instance as the characteristic polynomials of symplectic matrices. Using a suitably modified variant of the Fourier-analytic methods of Bhargava and others, we find that polynomials with non-generic Galois group appear with frequency O(log H/H) and, unlike in van der Waerden's setting, almost all of these polynomials are irreducible.

I will describe some connections between arithmetic geometry of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical invariants. Our formula extends a classical computation of local height functions due to Tate (involving Bernoulli polynomials).
Based on ongoing work with Robin de Jong.