Denote by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_{a,b,c,d,e}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</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> </mrow> </msub> </math></EquationSource> </InlineEquation> the algebraic curve over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> with affine equation given by <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} N_{a,b,c,d,e}: ax^4+bx^2+cy^2+dy+e=0. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>N</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> </mrow> </msub> <mo>:</mo> <mi>a</mi> <msup> <mi>x</mi> <mn>4</mn> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>e</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we find an expression for 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">$N_{a,b,c,d,e}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</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> </mrow> </msub> </math></EquationSource> </InlineEquation> in terms of p-adic <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-Appell series, extending a relation of the same with finite field <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-Appell series to all primes. In addition, we deduce certain transformation formulas for the p-adic <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$F_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>-Appell series analogous to their finite field and classical counterparts.

p-Adic $F_3$-Appell series and a family of algebraic curves

期刊论文
发表日期：

2026年3月31日

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Shaik Azharuddin,
Gautam Kalita

摘要

Denote by $N_{a,b,c,d,e}$ $N_{a, b, c, d, e}$ the algebraic curve over $\mathbb {Q}$ $Q$ with affine equation given by $\begin{aligned} N_{a,b,c,d,e}: ax^4+bx^2+cy^2+dy+e=0. \end{aligned}$ $\begin{matrix} N_{a, b, c, d, e} : a x^{4} + b x^{2} + c y^{2} + d y + e = 0 . \end{matrix}$ In this paper, we find an expression for the number of $\mathbb {F}_q$ $F_{q}$ -points on $N_{a,b,c,d,e}$ $N_{a, b, c, d, e}$ in terms of p-adic $F_3$ $F_{3}$ -Appell series, extending a relation of the same with finite field $F_3$ $F_{3}$ -Appell series to all primes. In addition, we deduce certain transformation formulas for the p-adic $F_3$ $F_{3}$ -Appell series analogous to their finite field and classical counterparts.

p-Adic \(F_3\)-Appell series and a family of algebraic curves

摘要