Distribution of rational points of an algebraic surface over finite fields
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912082445729792 |
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| author | Pujahari, Sudhir Saikia, Neelam |
| author_facet | Pujahari, Sudhir Saikia, Neelam |
| contents | The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(λ),$ where $A_p(λ)$ is a character sum and $λ$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(λ)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of rational points of an algebraic surface over finite fields Pujahari, Sudhir Saikia, Neelam Number Theory The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(λ),$ where $A_p(λ)$ is a character sum and $λ$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(λ)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields. |
| title | Distribution of rational points of an algebraic surface over finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.17340 |