The shifted convolution problem in function fields

Abstract: We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ where $f$ runs over monic polynomials in $\mathbb{F}q[T]$ of a given degree, and $h$ is a given monic polynomial. We prove an asymptotic formula in the range $\operatorname{deg}(h) < (2-\epsilon)\operatorname{deg}(f)$. We also consider mixed correlations and self-correlations of $r\chi = 1 \star \chi$, the convolution of $1$ with a Dirichlet character mod $\ell$, where $\ell$ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of $\mathbb{F}_q[T]$. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in $\mathbb{F}_q[T]$ which was not previously available.