Modular Curves Of Genus Research Articles

A single-variable proof of the omega SPT congruence family over powers of 5

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function omega (q). Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 {mathbb {Z}}[X]-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang’s family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of {mathbb {Z}}[X]. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.

Quadratic Chabauty for modular curves: algorithms and examples

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

Modular curves of prime-power level with infinitely many rational points

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.

Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

Let Y ns + ( n ) be the open non-cuspidal locus of the modular curve X ns + ( n ) associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on Y ns + ( n ) for suitably chosen n. In this note, we give a genus formula for the modular curves X ns + ( n ) and we give three new solutions to the class number one problem using the modular curves X ns + ( n ) for n = 16 , 20 , 21 . These are the only such modular curves of genus ⩽2 that had not yet been exploited.

Modular curves of genus 2

We prove that there is only a finite number of genus 2 curves C defined over Q such that there exists a nonconstant morphism pi:X_1(N) --->C defined over Q and the jacobian of C, J(C), is a Q-factor of the new part of the jacobian of X_1(N), J_1(N)^{new}. Moreover, we prove that there are only 149 genus two curves of this kind with the additional requeriment that their jacobians are Q-simple. We determine the corresponding newforms and present equations for all these curves.