Explicit moduli spaces for curves of genus 1 and 2

Abstract

In this thesis we study pairs of $N$-congruent elliptic curves. That is, we study pairs of elliptic curves whose $N$-torsion subgroups are isomorphic as Galois modules. In particular, we study the moduli surfaces $Z_{N,r}$ which parametrise $N$-congruences of elliptic curves which raise the Weil pairing to a power of $r$. We first study the birational geometry of the surfaces $Z_{N,r}^{\text{sym}}$ which arise as quotients of the surfaces $Z_{N,r}$ by the involution swapping the roles of the $N$-congruent elliptic curves. Building on previous work of Kani--Schanz and Hermann we compute the geometric genus of $Z_{N,r}^{\text{sym}}$ and place them within the Enriques--Kodaira classification in many cases. As a corollary we deduce that the Humbert surface of discriminant $N^2$ is rational if and only if $N \leq 16$, or if $N = 18$, $20$, or $24$. We then study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all, cases the modular curves in question correspond to the normaliser of a Cartan subgroup of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs, $(N,r)$, such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an example of a $48$-congruence over $\mathbb{Q}$. We make a conjecture classifying nontrivial $(N,r)$-congruences between quadratic twists of elliptic curves over $\mathbb{Q}$. We give a more detailed analysis of the level 15 case and use elliptic Chabauty to determine the rational points on a modular curve of genus 2 and Jacobian of rank $2$ which arises as a double cover of the modular curve $X(\mathrm{ns}, 3^+, \mathrm{ns}, 5^+)$. As a consequence we obtain a new proof of the class number 1 problem. We construct infinite families of pairs of (geometrically non-isogenous) $12$ and $14$-congruent elliptic curves defined over $\mathbb{Q}$. We also find two pairs of $15$-congruent elliptic curves over $\mathbb{Q}$. Our approach is to compute explicit birational models for the moduli surfaces $Z_{N,r}$ for each $N = 12$, $14$ and $15$. A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$, or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves. Finally, we study an application of congruences to visualising elements of the Tate--Shafarevich group of an abelian variety. In particular, we construct a number of genus 2 Jacobians $J/\mathbb{Q}$ with real multiplication by $\mathbb{Z}[\sqrt{2}]$ and whose Tate--Shafarevich groups contain an element of order $7$. Our approach is to find elliptic curves whose $7$-torsion subgroups are isomorphic as Galois modules to the $(3 \pm \sqrt{2})$-torsion subgroups of $J/\mathbb{Q}$.