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.
Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a “partially relative” symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a “higher-order” Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell–Weil group of its Jacobian.
Answering a question of Zureick-Brown, we determine the cubic points on the modular curves X 0 (N ) for N ∈ {53, 57, 61, 65, 67, 73} as well as the quartic points on X 0 (65). To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on X 0 (65), we rigorously compute the full rational Mordell-Weil group of its Jacobian.
We prove that every elliptic curve defined over a totally real number field of degree 4 not containing 5 \sqrt {5} is modular. To this end, we study the quartic points on four modular curves.
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