Generalized twisted Edwards curves over finite fields and hypergeometric functions Article Swipe
Rupam Barman
,
Sipra Mairty
,
Sulakashna
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2412.06199
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2412.06199
Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $a,b,c,d,e,f \in \mathbb{F}_q^{\times}$, denote by $C_{a,b,c,d,e,f}$ the family of algebraic curves over $\mathbb{F}_q$ given by the affine equation \begin{align*} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{align*} The family of generalized twisted Edwards curves is a subfamily of $C_{a,b,c,d,e,f}$. Let $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ denote the number of points on $C_{a,b,c,d,e,f}$ over $\mathbb{F}_q$. In this article, we find certain expressions for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ when $af=ce$. If $c^2-4ab\neq 0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of a $p$-adic hypergeometric function $\mathbb{G}(x)$ whose values are explicitly known for all $x\in \mathbb{F}_q$. Next, if $c^2-4ab=0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of another $p$-adic hypergeometric function and then relate it to the traces of Frobenius endomorphisms of a family of elliptic curves. Furthermore, using the known values of the hypergeometric functions, we deduce some nice formulas for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$.
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- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2412.06199
- https://arxiv.org/pdf/2412.06199
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4405254000
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https://openalex.org/W4405254000Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2412.06199Digital Object Identifier
- Title
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Generalized twisted Edwards curves over finite fields and hypergeometric functionsWork title
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preprintOpenAlex work type
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-12-09Full publication date if available
- Authors
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Rupam Barman, Sipra Mairty, SulakashnaList of authors in order
- Landing page
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https://arxiv.org/abs/2412.06199Publisher landing page
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https://arxiv.org/pdf/2412.06199Direct link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2412.06199Direct OA link when available
- Concepts
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Mathematics, Hypergeometric function, Pure mathematics, Hypergeometric distribution, Finite field, Generalized hypergeometric function, Basic hypergeometric series, Confluent hypergeometric function, Discrete mathematicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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