## Explicit moduli spaces for curves of genus 1 and 2

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## Abstract

In this thesis we study pairs of

We first study the birational geometry of the surfaces

We then study

We construct infinite families of pairs of (geometrically non-isogenous)

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