Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions Article Swipe

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Sulakashna , Rupam Barman · YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2307.11982

Let $D_λ^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^{d,k}:X_1^d+X_2^d=λdX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and $\gcd(d,k)=1$. Let $\#D^{d,k}_λ$ denote the number of points on $D_λ^{d,k}$ in $\mathbb{P}^{1}(\mathbb{F}_q)$. It is easy to see that $\#D_λ^{d,k}$ is equal to the number of distinct zeros of the polynomial $y^d-dλy^k+1\in \mathbb{F}_q[y]$ in $\mathbb{F}_q$. In this article, we prove that $\#D^{d,k}_λ$ is also equal to the number of distinct zeros of the polynomial $y^{d-k}(1-y)^k-(dλ)^{-d}$ in $\mathbb{F}_q$. We express the number of distinct zeros of the polynomial $y^{d-k}(1-y)^k-(dλ)^{-d}$ in terms of a $p$-adic hypergeometric function. Next, we derive summation identities for the $p$-adic hypergeometric functions appearing in the expressions for $\#D^{d,k}_λ$. Finally, as an application of the summation identities, we prove identities for the trace of Frobenius endomorphism on certain families of elliptic curves.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2307.11982

PDF

https://arxiv.org/pdf/2307.11982

OA Status

green

Related Works

10

OpenAlex ID

https://openalex.org/W4385245489

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OpenAlex ID

https://openalex.org/W4385245489

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DOI

https://doi.org/10.48550/arxiv.2307.11982

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Title

Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions

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Type

preprint

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Language

en

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Publication year

2023

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Publication date

2023-07-22

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Authors

Sulakashna, Rupam Barman

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https://arxiv.org/abs/2307.11982

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https://arxiv.org/pdf/2307.11982

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Yes

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OA status

green

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https://arxiv.org/pdf/2307.11982

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Concepts

Lambda, Hypergeometric function, Mathematics, Polynomial, Combinatorics, Diagonal, Finite field, Hypersurface, Physics, Mathematical analysis, Geometry, Optics

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0

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Related works (count)

10

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