New formulas for Faltings delta invariant

Abstract: In this talk we will give new formulas for Faltings delta invariant of Riemann surfaces. This invariant plays a crucial role in Arakelov theory of arithmetic surfaces, where it appears in the arithmetic Noether formula. We will deduce a lot of applications, for example, a lower bound of the delta invariant only in terms of the genus, a canonical extension of the delta invariant to abelian varieties, an improved version of Szpiro's small points conjecture for cycle covers, a bound of the Arakelov heights of Weierstraß points in terms of the Faltings height and a bound of the Arakelov-Green function in terms of the delta invariant. In the case of hyperelliptic curves we will also obtain an arithmetic analogous of the Bogomolov-Miyaoka-Yau inequality and an effective version of the Bogomolov conjecture.