Heights on algebraic curves
Tony Shaska, Lubjana Beshaj · NATO science for peace and security series. D, Information and communication security · 2015
In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves ${\mathfrak{H} (f) \le c_0 \tilde{H} (f)}$ where c0is a constant and $\tilde{H}$ the minimal height of the corresponding binary form. For genus g = 2 and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.