Title: The shifted convolution problem in function fields
Abstract: We will discuss some results on the shifted convolution problem for the divisor function over function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ where $f$ runs over monic polynomials of given degree in $\mathbb{F}_q[T]$, and $h$ is a given monic polynomial. We prove an asymptotic formula in the range $\deg(h) < (2-\epsilon)\deg(f)$. The central ingredient for this work is a Voronoi summation formula for the divisor function. The results also extend to various correlations of the convolution of $1$ with a Dirichlet character mod $\ell$, where $\ell$ is a monic irreducible polynomial. This is joint work with Alexandra Florea, Amita Malik, and Anurag Sahay.