Arithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However... Show moreArithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However, it is usually very difficult to answer questions like this.A. Beilinson and S. Bloch conjectured a very general height theory in 1980s, which was used by B. Gross and R. Schoen in their study of the Gross-Schoen cycles. The height of canonical Gross-Schoen cycles is conjectured to be non-negative. This was verified when the curve is an elliptic or hyperelliptic curve, while very few are known in the non-hyperelliptic case.During my PhD study, I study the Beilinson-Bloch height of canonical Gross-Schoen cycles on curves with an emphasis on the genus 3 case (almost all genus 3 curves are non-hyperelliptic). I studied its unboundedness and singular properties, and did explicit computation for the height of the canonical Gross-Schoen cycle of a specific plane quartic curve.The method used in my thesis should be helpful for verifications. Show less
We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the Néron-Tate... Show moreWe use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the Néron-Tate height of the corresponding point on the Jacobian. We give an algorithm to compute the set of points of bounded height with respect to this new height. This provides an `in principle' solution to the problem of determining the sets of points of bounded Néron-Tate heights on the Jacobian. We give a worked example of how to compute the bound over a global function field for several curves, of genera up to 11. 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
