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Quadratic points on modular curves with infinite Mordell--Weil group | | Josha Box
; | | Date: |
12 Jun 2019 | | 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 $mathbb{Q}$-curves or have
complex multiplication. | | Source: | arXiv, 1906.5206 | | Services: | Forum | Review | PDF | Favorites | |
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