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#### The delta invariant in Arakelov geometry

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https://hdl.handle.net/20.500.11811/6757

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##### Citation

Wilms, R. (2016). The delta invariant in Arakelov geometry. PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.

Cite as: https://hdl.handle.net/21.11116/0000-0004-10DC-2

##### Abstract

In this thesis we study Faltings' delta invariant of compact and connected Riemann surfaces.

This invariant plays a crucial role in Arakelov theory of arithmetic surfaces. For example, it appears in the arithmetic Noether formula. We give new explicit formulas for the delta invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for it only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of the delta invariant. Furthermore, we give a canonical extension of Faltings' delta invariant to the moduli space of indecomposable principally polarised complex abelian varieties. As applications to Arakelov theory, we obtain bounds for the Arakelov heights of the Weierstraß points and for the Arakelov intersection number of any geometric point with certain torsion line bundles in terms of the Faltings height. Moreover, we deduce an improved version of Szpiro's small points conjecture for cyclic covers of prime degree and an explicit expression for the Arakelov self-intersection number of the relative dualizing sheaf, an effective version of the Bogomolov conjecture and an arithmetic analogue of the Bogomolov-Miyaoka-Yau inequality for hyperelliptic curves.

This invariant plays a crucial role in Arakelov theory of arithmetic surfaces. For example, it appears in the arithmetic Noether formula. We give new explicit formulas for the delta invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for it only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of the delta invariant. Furthermore, we give a canonical extension of Faltings' delta invariant to the moduli space of indecomposable principally polarised complex abelian varieties. As applications to Arakelov theory, we obtain bounds for the Arakelov heights of the Weierstraß points and for the Arakelov intersection number of any geometric point with certain torsion line bundles in terms of the Faltings height. Moreover, we deduce an improved version of Szpiro's small points conjecture for cyclic covers of prime degree and an explicit expression for the Arakelov self-intersection number of the relative dualizing sheaf, an effective version of the Bogomolov conjecture and an arithmetic analogue of the Bogomolov-Miyaoka-Yau inequality for hyperelliptic curves.