Quadratic points on modular curves with infinite Mordell–Weil group

DOI10.1090/MCOM/3547zbMATH Open1462.11049arXiv1906.05206OpenAlexW3012464010MaRDI QIDQ5131005

Author name not available (Why is that?)

Publication date: 31 October 2020

Published in: (Search for Journal in Brave)

Abstract: Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)o{X}_0(N{)}^{+}$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $X_0(N{)}^{+}$, and we specify the exceptional quadratic points on $X_0(N)$ not coming from $X_0(N{)}^{+}$. In particular we determine the $j$-invariants of the corresponding elliptic curves and whether they are $mathbbQ$-curves or have complex multiplication.

Full work available at URL: https://arxiv.org/abs/1906.05206

zbMATH Keywords

No records found.

Mathematics Subject Classification ID

No records found.

This page was built for publication: Quadratic points on modular curves with infinite Mordell–Weil group

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q5131005)