We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d(f) d(f+h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where f runs over monic polynomials in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}_q[T]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a given degree, and h is a given monic polynomial. We prove an asymptotic formula in the range <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\deg (h) < (2-\epsilon )\deg (f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>deg</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo><</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>-</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>deg</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also consider mixed correlations and self-correlations of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r_\chi = 1 \star \chi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mi>χ</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>⋆</mo> <mi>χ</mi> </mrow> </math></EquationSource> </InlineEquation>, the convolution of 1 with a Dirichlet character mod <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {F}_q[T]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {F}_q[T]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which was not previously available.

The shifted convolution problem in function fields

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2026年3月19日

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Alexandra Florea,
Matilde Lalín,
Amita Malik,
Anurag Sahay

摘要

We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ $d (f) d (f + h)$ where f runs over monic polynomials in $\mathbb {F}_q[T]$ $F_{q} [T]$ of a given degree, and h is a given monic polynomial. We prove an asymptotic formula in the range $\deg (h) < (2-\epsilon )\deg (f)$ $deg (h) < (2 - ϵ) deg (f)$ . We also consider mixed correlations and self-correlations of $r_\chi = 1 \star \chi $ $r_{χ} = 1 ⋆ χ$ , the convolution of 1 with a Dirichlet character mod $\ell $ $ℓ$ , where $\ell $ $ℓ$ 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 $\mathbb {F}_q[T]$ $F_{q} [T]$ . A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in $\mathbb {F}_q[T]$ $F_{q} [T]$ which was not previously available.