Exponential sums over singular binary quartic forms and applications
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917627020967936 |
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| author | Ishitsuka, Yasuhiro Taniguchi, Takashi Thorne, Frank Xiao, Stanley Yao |
| author_facet | Ishitsuka, Yasuhiro Taniguchi, Takashi Thorne, Frank Xiao, Stanley Yao |
| contents | We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector spaces, but also exhibits a striking new feature: the point counting function $a_p(E) = p + 1 - \#E(\mathbb{F}_p)$ associated to an associated elliptic curve makes a prominent appearance. The proof techniques are also new, involving techniques from elementary algebraic geometry and classical invariant theory. As an application to prime number theory, we demonstrate the existence of `many' 2-Selmer elements for elliptic curves with discriminants that are squarefree and have at most four prime factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential sums over singular binary quartic forms and applications Ishitsuka, Yasuhiro Taniguchi, Takashi Thorne, Frank Xiao, Stanley Yao Number Theory We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector spaces, but also exhibits a striking new feature: the point counting function $a_p(E) = p + 1 - \#E(\mathbb{F}_p)$ associated to an associated elliptic curve makes a prominent appearance. The proof techniques are also new, involving techniques from elementary algebraic geometry and classical invariant theory. As an application to prime number theory, we demonstrate the existence of `many' 2-Selmer elements for elliptic curves with discriminants that are squarefree and have at most four prime factors. |
| title | Exponential sums over singular binary quartic forms and applications |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.00541 |