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We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture,over a function field of characteristic 0 by proving Zhang's Conjecture aboutcertain invariants of metrized graphs. In the function field case, theseconjectures were previously known to be true only for curves of genus at most 4and a few other special cases. We also either verify or improve the previousresults. We relate the invariants involved in Zhang's Conjecture to the tauconstant of metrized graphs. Then we use and extend our previous results on thetau constant. By proving another Conjecture of Zhang, we obtain a new proof ofthe slope inequality for Faltings heights on moduli space of curves.

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields