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arXiv · 2310.10432

Quadratic points on $X_0(163)$

Philippe Michaud-Jacobs·Filip Najman
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Abstract

We determine all the quadratic points on the genus $13$ modular curve $X_0(163)$, thus completing the answer to a recent question of Banwait, the second-named author, and Padurariu. In doing so, we investigate a curious phenomenon involving a cubic point with complex multiplication on the curve $X_0(163)$. This cubic point prevents us, due to computational restraints, from directly applying the state-of-the-art Atkin--Lehner sieve for computing quadratic points on modular curves $X_0(N)$. To overcome this issue, we introduce a technique which allows us to work with the Jacobian of curves modulo primes by directly computing linear equivalence relations between divisors.

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Philippe Michaud-Jacobs, Filip Najman. 2023-10-16. Quadratic points on $X_0(163)$. https://arxiv.org/abs/2310.10432

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