Harron and Snowden (J. reine angew. Math. 729 (2017), 151-170) counted the number of elliptic curves over Q$\mathbb {Q}$ up to height X$X$ with torsion group G$G$ for each possible torsion group G... Show moreHarron and Snowden (J. reine angew. Math. 729 (2017), 151-170) counted the number of elliptic curves over Q$\mathbb {Q}$ up to height X$X$ with torsion group G$G$ for each possible torsion group G$G$ over Q$\mathbb {Q}$. In this paper, we generalize their result to all number fields and all level structures G$G$ such that the corresponding modular curve XG$X_G$ is a weighted projective line P(w0,w1)$\mathbb {P}(w_0,w_1)$ and the morphism XG -> X(1)$X_G\rightarrow X(1)$ satisfies a certain condition. In particular, this includes all modular curves X1(m,n)$X_1(m,n)$ with coarse moduli space of genus 0. We prove our results by defining a size function on P(w0,w1)$\mathbb {P}(w_0,w_1)$ following unpublished work of Deng (Preprint, ), and working out how to count the number of points on P(w0,w1)$\mathbb {P}(w_0,w_1)$ up to size X$X$. Show less
This thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are... Show moreThis thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are examples of arithmetic surfaces. Therefore, Arakelov theory (intersection theory on arithmetic surfaces) occupies a prominent place in this thesis. Apart from this, a substantial part of it is devoted to studying modular curves over finite fields, and their Jacobian varieties, from an algorithmic viewpoint. The end product of this thesis is an algorithm for computing modular Galois representations. These are certain two-dimensional representations of the absolute Galois group of the rational numbers that are attached to Hecke eigenforms over finite fields. The running time of our algorithm is (under minor restrictions) polynomial in the length of the input. This main result is a generalisation of that of work of Jean-Marc Couveignes, Bas Edixhoven et al. Several intermediate results are developed in sufficient generality to make them of interest to the study of modular curves and modular forms in a wider sense. Show less
