Circle Method and Related Topics

We will discuss ongoing work with Alexandra Florea, Matilde Lalín, and Amita Malik on the shifted convolution problem for divisor functions in function fields. This involves studying the average value of $d(f) d(f+h)$ where $h$ is a fixed polynomial (having possibly large degree $m$) in $\mathbb{F}_q[T]$ and $f$ runs over all monic polynomials in $\mathbb{F}_q[T]$ of degree $n$, where $n$ goes to infinity. Our techniques mirror the classical approach of Estermann in the integer setting. The main new ingredient is a functional equation for the Estermann function (equivalently, a Voronoi summation formula for the divisor function) that was not previously available in function fields. If time permits, we will discuss a related result involving the shifted convolution of the norm-counting functions of quadratic extensions. The talk should be accessible to those unfamiliar with function fields.

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive. The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to consider variants of the problem where the Eisenstein series is replaced by a cusp form. I will discuss recent work with Agniva Dasgupta and Wing Hong Leung where we obtain a nontrivial bound on this second moment. I will also discuss some applications, including an improvement on the Rankin-Selberg problem.

We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This would allow for example asymptotic formulas for the density of solutions to nonsingular systems of Diophantine inequalities in sufficiently many variables.

I will discuss the problem of obtaining an asymptotic formula for counting integral solutions to an equation of the form f(x, y, z, w)=N in an expanding box, where N is a non-zero integer and f is an indefinite quadratic form over the integers. For forms of the shape axy-bzw, I will then explain how we can obtain a significantly strong error term by applying deep methods from the spectral theory of automorphic forms. This is a joint work with Rachita Guria.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon L. Rydin Myerson (warwick) and Junxian Li (UC Davis).

n this talk, I will discuss the problem of obtaining lower bounds for the number of rational points of bounded height on cubic hypersufaces. Our main tools will be the circle method, the Ekedahl sieve and the geometry of numbers.

In practice, L-functions appear as generating functions encapsulating information about various objects, such as Galois representations, elliptic curves, arithmetic functions, modular forms, Maass forms, etc. Studying L-functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We discuss a connection between low-lying zeros and central values of L-functions, in particular showing that results about the distribution of low-lying zeros (towards the density conjecture of Katz-Sarnak) implies results about the distribution of the central values (towards the normal distribution conjecture of Keating-Snaith). Even though we discuss this principle in general, we instanciate it in the case of modular forms in the level aspect to give a statement an...