In this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major... Show moreIn this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major results include the polynomiality of the double ramification cycle and recursive relations for the log double ramification cycle.In the second half we study rational points on curves, in particular Chabauty's method for finding the rational points and extensions of it. Major results include that the geometric (quadratic) Chabauty method is theoretically stronger than the original (quadratic) Chabauty method, and that local heights for quadratic Chabauty are explicitly computable. Show less
Chapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the... Show moreChapter 1,contains the numerical verification of the Birch and Swinnerton-Dyer conjecture for hundreds of Jacobians of hyperelliptic curves of genus 2, 3, 4 and 5. Chapter 2 treats the equivalence of BSD for a certain elliptic curve over Q(∜5), and a pair of hyperelliptic curves over Q. In chapter 3, ordinary Galois covers of smooth curves are constructed from ordinary Galois covers of semi-stable curves. Finally, in chapter 4, the statistics of ordinary reduction for hyperelliptic curves is considered. Show less
