On the Distributions of Point Counts on Hypergeometric Varieties

This thesis is on arithmetic statistics that arise from arithmetic geometry. In particular, I prove arithmetic-statistical results about the distributions of points on hypergeometric varieties. As modular forms are central objects in these results, I provide a new proof of the celebrated Eichler--Selberg trace formula for levels dividing 4. The central elements of this paper are elliptic curves, finite field hypergeometric functions, harmonic Maass forms, and holomorphic modular forms.

It is natural to study statistical questions about the number of finite-field points on algebraic varieties. One of the most famous such questions is the Sato--Tate conjecture on the distribution of the traces of Frobenius for a fixed non-CM elliptic curves as one goes over all primes of good reduction. In their landmark work, Richard Taylor and his collaborators proved this conjecture using deep tools from the analytic theory of automorphic forms, l-adic Galois representations and étale cohomology. Motivated by this conjecture, in the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the Sato--Tate distribution. Inspired by the deep result of Taylor and the seminal work of Birch, I determine the limiting distribution of Frobenius traces for the family of Legendre elliptic curves and a special family of K3 surfaces. In order to prove these results, in joint work with Saikia and Ono, I use results from arithmetic geometry to compute the moments of the traces in terms of Hurwitz class numbers, apply the theory of holomorphic modular forms and harmonic Maass forms to determine the asymptotics of those moments, and then apply the method of moments to deduce the limiting distribution.

Motivated by the work of Birch, Murty and Prabhu make Birch's results effective by asymptotically bounding the error term. In light of the fact that the limiting distribution for the K3 surfaces has vertical asymptotics, and inspired by Murty and Prabhu's work, I explicitly bound the error in the limiting distribution for K3 surfaces. In order to do so, I use Rankin--Selberg unfolding to explicitly write the modular forms that encode the moment formulas in terms of newforms, apply Deligne's theorem that bounds Fourier coefficients of newforms, and then use Beurling--Selberg polynomials to determine distributions from moments explicitly.

The Eichler--Selberg trace formulas express the traces of Hecke operators on a
spaces of cusp forms in terms of weighted sums of Hurwitz--Kronecker class numbers. For
cusp forms on SL_2(Z), Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. Furthermore, the trace formulas for SL_2(Z) play an essential role in Birch's result. Using Zagier's method, in joint work with Ono, I prove the Eichler--Selberg trace formulas for Gamma_0(2) and Gamma_0(4). To do this, I use Zagier's Eisenstein series, Rankin--Selberg unfolding, the Petersson inner product, and the theory of holomorphic modular forms.

In their famous work, Feit and Fine count the number of pairs of commuting nxn matrices with entries in a finite field. This can be framed as counting F_p-points on the commuting variety defined by (A,B) of nxn matrices which satisfy the equation AB-BA = 0_n. The geometry of this variety, usually called the commuting variety, was studied in deep work of Motzkin and Taussky and Gerstenhaber. Motivated by this, in joint work with Huang and Ono, I count the number of nxn matrix points on Legendre elliptic curves and K3 surfaces in terms of finite field hypergeometric functions and partitions. Using this explicit point count, I determine the limiting distribution of the ``random part'' of the matrix point counts as the finite field grows. In order to do this, I use results of Huang that connect the number of matrix points to the zeta function of a variety.