On Faltings' Delta-Invariant of Hyperelliptic Riemann Surfaces

2015/05/18 by Wilms, Robert
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.04740

Abstract

In this paper we prove new explicit formulas for Faltings' δ-invariant of an arbitrary hyperelliptic Riemann surface. This has several applications: For example we obtain an explicit lower bound for δ depending only on the genus, and we deduce new explicit bounds for the Arakelov self-intersection number ω² 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 θ-derivatives conjectured by Guàrdia.

Related

• Faltings' delta-invariant of a hyperelliptic Riemann surface
• On Szpiro's Discriminant Conjecture
• Effective bounds for Faltings's delta function
• Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects
• The a-number of hyperelliptic curves