The shifted convolution problem in function fields

TL;DR

The paper advances the analytic theory of shifted convolution in function fields by establishing an asymptotic formula for $\frac{1}{q^n}\sum_{f\in \mathcal{M}_n} d(f)d(f+h)$ in the large degree limit, valid when $\deg(h)<(2-\varepsilon)\deg(f)$. A central tool is a novel Voronoi summation scheme in $\mathbb{F}_q[T]$ derived from functional equations for function-field analogues of Hurwitz and Estermann functions, including character twists. The authors extend these methods to mixed correlations and self-correlations of $r_\chi=1\star\chi$, obtaining asymptotics in various regimes and uncovering a rich structure for quadratic characters that connects to norm-counting functions of quadratic extensions. This framework integrates Kloosterman sum bounds, Linnik–Selberg cancellation results, and Perron-type formulae to produce explicit main terms and sharp error estimates, broadening the scope of function-field analogues of the shifted convolution problem. The results have implications for understanding moments and correlation phenomena in function fields and provide a robust set of tools for related arithmetic questions in finite 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 $\operatorname{deg}(h) < (2-ε)\operatorname{deg}(f)$. We also consider mixed correlations and self-correlations of $r_χ= 1 \star χ$, 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.

Theorems & Definitions (52)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 2.1
• Remark 2.2