Let <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> be a finite field with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q=p^r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>r</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> elements. For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a,b,c,d,e,f \in \mathbb {F}_q^{\times }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mo>×</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, consider the family of plane affine curves over <InlineEquation ID="IEq4"> <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> defined by <Equation ID="Equ51"> <EquationSource Format="TEX">$\begin{aligned} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>:</mo> <mi>a</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>f</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mi>y</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Denote by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the number of <InlineEquation ID="IEq6"> <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>-points on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_{a,b,c,d,e,f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. In this article, we obtain explicit formulas for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under the condition <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$af=ce$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>f</mi> <mo>=</mo> <mi>c</mi> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$c^2-4ab\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we express <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of a p-adic hypergeometric function <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {G}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, whose values are explicitly known for all <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$x\in \mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$c^2-4ab=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we express <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of a different p-adic hypergeometric function and relate this expression to the traces of Frobenius endomorphisms of a family of elliptic curves. Finally, by using known evaluations of these hypergeometric functions, we deduce some nice formulas for <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mi>C</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Generalized twisted Edwards curves over finite fields and hypergeometric functions

期刊论文
发表日期：

2026年7月9日

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Rupam Barman,
Sipra Maity,
Sulakashna

摘要

Let $\mathbb {F}_q$ $F_{q}$ be a finite field with $q=p^r$ $q = p^{r}$ elements. For $a,b,c,d,e,f \in \mathbb {F}_q^{\times }$ $a, b, c, d, e, f \in F_{q}^{\times}$ , consider the family of plane affine curves over $\mathbb {F}_q$ $F_{q}$ defined by $\begin{aligned} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{aligned}$ $\begin{matrix} C_{a, b, c, d, e, f} : a y^{2} + b x^{2} + c x y = d + e x^{2} y^{2} + f x^{3} y . \end{matrix}$ Denote by $\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$ $# C_{a, b, c, d, e, f} (F_{q})$ the number of $\mathbb {F}_q$ $F_{q}$ -points on $C_{a,b,c,d,e,f}$ $C_{a, b, c, d, e, f}$ . In this article, we obtain explicit formulas for $\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$ $# C_{a, b, c, d, e, f} (F_{q})$ under the condition $af=ce$ $a f = c e$ . When $c^2-4ab\ne 0$ $c^{2} - 4 a b \neq 0$ , we express $\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$ $# C_{a, b, c, d, e, f} (F_{q})$ in terms of a p-adic hypergeometric function $\mathbb {G}(x)$ $G (x)$ , whose values are explicitly known for all $x\in \mathbb {F}_q$ $x \in F_{q}$ . When $c^2-4ab=0$ $c^{2} - 4 a b = 0$ , we express $\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$ $# C_{a, b, c, d, e, f} (F_{q})$ in terms of a different p-adic hypergeometric function and relate this expression to the traces of Frobenius endomorphisms of a family of elliptic curves. Finally, by using known evaluations of these hypergeometric functions, we deduce some nice formulas for $\#C_{a,b,c,d,e,f}(\mathbb {F}_q)$ $# C_{a, b, c, d, e, f} (F_{q})$ .