Quadratic points on modular curves with infinite Mordell–Weil group

Abstract

Bruin and Najman [5] and Ozman and Siksek [33] have recently determined the quadratic points on each modular curve X 0 (N ) of genus 2, 3, 4, or 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 ) → 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 Q-curves or have complex multiplication. 1 2 JOSHA BOX for N ≥ B d are cuspidal. Moreover, for prime values of N , Merel's uniform boundedness theorem [30] proves the existence of such an upper bound B d for each d.The low degree points on X 0 (N ), however, are naturally more abundant than on X 1 (N ), which complicates their study.There are two obvious potential sources of infinitely many quadratic points on X 0 (N ): a degree 2 map X 0 (N ) → P 1 over Q (which exists if and only if X 0 (N ) is hyperelliptic) and a degree 2 map over Q to an elliptic curve with infinite Mordell-Weil group (the existence of which implies that X 0 (N ) is bielliptic). In both of these cases the rational points on the image give rise to infinitely many quadratic points on X 0 (N ). Using Faltings' theorem [13] on abelian varieties, Abramovich and Harris [1] show that the set of quadratic points on X 0 (N ) is infinite in these two cases only. Building on the work of Harris and Silverman [15], Bars [4] then determined all bielliptic X 0 (N ) and decided that exactly 10 of these have a quotient of positive Mordell-Weil rank. Moreover, Ogg [31] decided for which 19 values of N the curve X 0 (N ) is hyperelliptic. Consequently, X 0 (N ) has infinitely many quadratic points for the following 28 explicit values of N 131. The careful reader will have noticed there must be one curve that is both hyperelliptic and bielliptic with quotient of positive rank: this is the infamous X 0 (37).Even when infinite, the quadratic points on X 0 (N ) can be described. Recently, Bruin and Najman [5] determined the finitely many exceptional quadratic points (those not coming from P 1 (Q)) on all hyperelliptic X 0 (N ) except for N = 37, and they moreover proved in those cases that the quadratic points coming from P 1 (Q) correspond to Q-curves. The remaining case N = 37 is also the only hyperelliptic one where J 0 (N )(Q) is infinite. Subsequently, Ozman and Siksek [33] determined all finitely many quadratic points on X 0 (N ) when it is non-hyperelliptic of genus 2, 3, 4 or 5 and the Mordell-Weil group J 0 (N )(Q) is finite.

Quadratic points on bielliptic modular curves

“…We will be building upon the work of Box in [4], which in turns builds on the work of Siksek [29]. In [4, Theorem 2.1.]…”

“…Box [4] completed the description of quadratic points on X 0 (n) of genus 2 ≤ g(X 0 (n)) ≤ 5 by describing the quadratic points for the 8 values of n such that rk(J 0 (n)(Q)) > 0, including the hyperelliptic case of n = 37. In the cases n = 43, 53, 61, the curves X 0 (n) turn out to be bielliptic with a bielliptic map b : X 0 (n) → X + 0 (n) and such that X + 0 (n) is an elliptic curve of positive rank over Q.…”

“…Bars [3] determined all the bielliptic modular curves X 0 (n) (there are 41 of them) and those among them with infinitely many quadratic points (28 out of the 41 satisfy this). Since many of the bielliptic curves have genus ≤ 5, the quadratic points on all but 12 have already been described in the aforementioned papers [6,28,4]. In the table below we list the remaining values n, their genus g(X 0 (n)), and the rank rk(J 0 (n)(Q)) of their Jacobian over Q.…”

“…The second method, used to deal with n ∈ {60, 79, 83, 89, 95, 101, 119, 131} is, following Siksek [29] and Box [4], the symmetric relative Chabauty method. In principle what we do in these cases is very similar to [4], but there is a number of tweaks and minor improvements that we do to make the necessary computations, which have previously been done on curves of genus only up to 5, work on our curves, which are of genus 6 ≤ g ≤ 11. We explain these…”

“…

Bruin and Najman [6], Ozman and Siksek [28], and Box [4] described all the quadratic points on the modular curves of genus 2 ≤ g(X 0 (n)) ≤ 5. Since all the hyperelliptic curves X 0 (n) are of genus ≤ 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 (n) naturally arises; this question has recently also been posed by Mazur.We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves X 0 (n) for which this has not been done already.

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