Heights on algebraic curves

Abstract

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 $\mathcal{H}(f)\le c_0\tilde{H}(f)$ where $c_0$ is 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.

Fixes some minor typos and corrected some missing references from the first version. Information security, coding theory and related combinatorics, 165-202, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS, Amsterdam, 2015

References in corpus (7)

• Some special families of hyperelliptic curves
• Determining equations of families of cyclic curves
• Thetanulls of cyclic curves of small genus
• Bielliptic curves of genus 3 in the hyperelliptic moduli
• On Jacobians of curves with superelliptic components
• Families of genus two curves with many elliptic subcovers
• Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians