The shifted convolution problem in function fields

TL;DR

This paper investigates a mathematical problem involving polynomial divisor functions in function fields, proving asymptotic formulas and introducing a new Voronoi summation formula.

Contribution

The paper introduces a novel Voronoi summation formula in function fields, enabling new asymptotic results for shifted convolution problems.

Findings

An asymptotic formula is proven for the shifted convolution problem in the range deg(h) < (2-ε)deg(f).

Asymptotic results are derived for mixed and self-correlations involving Dirichlet characters.

The Voronoi summation formula is applied to study norm-counting functions in quadratic extensions of F_q[T].

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 \documentclass[12pt]{minimal}
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\begin{document}$$d(f) d(f+h)$$\end{document}d(f)d(f+h) where f runs over monic polynomials in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_q[T]$$\end{document}Fq[T] of a given degree, and h is a given monic polynomial. We prove an asymptotic formula in the range \documentclass[12pt]{minimal}
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\begin{document}$$\deg (h) < (2-\epsilon )\deg (f)$$\end{document}deg(h)<(2-ϵ)deg(f). We also consider mixed correlations and self-correlations of \documentclass[12pt]{minimal}
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\begin{document}$$r_\chi = 1 \star \chi $$\end{document}rχ=1⋆χ, the convolution of 1 with a Dirichlet character mod \documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}ℓ, where \documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}ℓ 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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_q[T]$$\end{document}Fq[T]. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_q[T]$$\end{document}Fq[T] which was not previously available.