2019/06/12 by Box, Josha · 2 citations
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1906.05206
Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves X0(N) of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the X0(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 X0(N)→ X0(N)+ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on X0(N)+, and we specify the exceptional quadratic points on X0(N) not coming from X0(N)+. In particular we determine the j-invariants of the corresponding elliptic curves and whether they are ℚ-curves or have complex multiplication.