Admissible Invariants of genus 3 Curves

Abstract

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus $3$. We find the sharp lower bound for the invariants $\varphi$, $\lambda$ and $\epsilon$ for all polarized metrized graphs of genus $3$. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.