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Quadratic points on $X_0(163)$ by Michaud-Jacobs, Philippe; Najman, Filip is a scholarly article available to read on EtoBox.

What is Quadratic points on $X_0(163)$ about?

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.

Author

Michaud-Jacobs, Philippe; Najman, Filip

Published

2023

Language

EN