Let E be an elliptic curve over the finite field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {F}}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> of q elements, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$P \in E({\mathbb {F}}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {F}}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-rational point. We study the sums <Equation ID="Equ13"> <EquationSource Format="TEX">$ S_{\chi ,P}(N,h) = \sum _{n=1}^N \chi (\psi _n(P)) \chi (\psi _{n+h}(P)), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>S</mi> <mrow> <mi>χ</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mi>h</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\psi _n(P)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the n-th division polynomial evaluated at P, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> is a multiplicative character of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\mathbb {F}}_q^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>. We estimate <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S_{\chi ,P}(N,h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mrow> <mi>χ</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on average over h over a rather short interval <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h \in [1, H]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>H</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We also obtain a multidimensional generalisation of this result.

On the correlations between character sums of division polynomials under shifts

期刊论文
发表日期：

2025年12月5日

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Subham Bhakta,
Igor E. Shparlinski

摘要

Let E be an elliptic curve over the finite field ${\mathbb {F}}_q$ $F_{q}$ of q elements, and $P \in E({\mathbb {F}}_q)$ $P \in E (F_{q})$ be an ${\mathbb {F}}_q$ $F_{q}$ -rational point. We study the sums $ S_{\chi ,P}(N,h) = \sum _{n=1}^N \chi (\psi _n(P)) \chi (\psi _{n+h}(P)), $ $S_{χ, P} (N, h) = \sum_{n = 1}^{N} χ (ψ_{n} (P)) χ (ψ_{n + h} (P)),$ where $\psi _n(P)$ $ψ_{n} (P)$ denotes the n-th division polynomial evaluated at P, and $\chi $ $χ$ is a multiplicative character of ${\mathbb {F}}_q^{*}$ $F_{q}^{*}$ . We estimate $S_{\chi ,P}(N,h)$ $S_{χ, P} (N, h)$ on average over h over a rather short interval $h \in [1, H]$ $h \in [1, H]$ . We also obtain a multidimensional generalisation of this result.