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arXiv · 2402.05599

A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture

Masahito Hayashi·Kazuyasu Shigemoto·Takuya Tsukioka
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Abstract

For quadratic curves over $F_p$, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a quadratic curve analogue of the modular form in the Taniyama-Shimura conjecture is proposed. This modular form yields the Gaussian sum and also possesses some modular transformation structure.

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Masahito Hayashi, Kazuyasu Shigemoto, Takuya Tsukioka. 2024-09-23. A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture. https://doi.org/10.24018/ejmath.2024.5.6.378

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