February 8, 2024Open Access

A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture

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Abstract

For quadratic curves over Q in Fₚ, 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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Hayashi et al. (Thu,) studied this question.

synapsesocial.com/papers/68e7b940b6db64358770fa5e https://doi.org/https://doi.org/10.48550/arxiv.2402.05599

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