On Faltings' Delta-Invariant of Hyperelliptic Riemann Surfaces

In this paper we prove new explicit formulas for Faltings' $\delta$-invariant of an arbitrary hyperelliptic Riemann surface. This has several applications: For example we obtain an explicit lower bound for $\delta$ depending only on the genus, and we deduce new explicit bounds for the Arakelov self-intersection number $\omega^2$ associated to hyperelliptic curves over number fields. Furthermore, we obtain an improved version of Szpiro's small points conjecture for hyperelliptic curves of genus at least $3$. Our method allows us in addition to establish a generalization of Rosenhain's formula on $\theta$-derivatives conjectured by Gu\`ardia.