On Faltings' Delta-Invariant of Hyperelliptic Riemann Surfaces

Abstract

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 Guàrdia.

arXiv commentThis paper is out-of-date. The results are completely contained in the newer paper: arXiv:1605.00847

References in corpus (1)

• Johnson's homomorphisms and the Arakelov-Green function