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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.18757 |
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| _version_ | 1866909806231552000 |
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| author | Hundley, Joseph Vega, Yaniel Rivera Scharaschkin, Victor |
| author_facet | Hundley, Joseph Vega, Yaniel Rivera Scharaschkin, Victor |
| contents | Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field $F$, when either $F$ contains three third roots of unity, or the defining polynomial is reducible. The restriction on $F$ allowed them, among other things, to reduce the case of irreducible polynomials of the form $x^3-a$. Pleso (2009) began the work of removing the restriction on $F$ by expressing the integral as a sum of $16$ integrals for the cubic polynomial $x^3 - b x - c$ with $b,c\in F$, and computing nine of them. In this work, we compute $15$ of Pleso's integrals, and reduce the last to an elementary assertion about the number of points on a surface over a finite field, in the special case when $F$ is the $p$-adic numbers, $F=\mathbb{Q}_p$, and $p$ is equivalent to $5$ mod $6$. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials, in a higher-rank setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18757 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Theorem of Jiang and Rallis Hundley, Joseph Vega, Yaniel Rivera Scharaschkin, Victor Number Theory 11S23 Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field $F$, when either $F$ contains three third roots of unity, or the defining polynomial is reducible. The restriction on $F$ allowed them, among other things, to reduce the case of irreducible polynomials of the form $x^3-a$. Pleso (2009) began the work of removing the restriction on $F$ by expressing the integral as a sum of $16$ integrals for the cubic polynomial $x^3 - b x - c$ with $b,c\in F$, and computing nine of them. In this work, we compute $15$ of Pleso's integrals, and reduce the last to an elementary assertion about the number of points on a surface over a finite field, in the special case when $F$ is the $p$-adic numbers, $F=\mathbb{Q}_p$, and $p$ is equivalent to $5$ mod $6$. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials, in a higher-rank setting. |
| title | On a Theorem of Jiang and Rallis |
| topic | Number Theory 11S23 |
| url | https://arxiv.org/abs/2507.18757 |