We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group SL 2 (Z) constitute a subalgebra of MD this also yields linear relations in MD. Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those in length 2, coming from modular forms.

No abstract

We present an arithmetic intersection theory for hermitian line bundles on arithmetic surfaces, where the metrics are allowed to have logarithmic singularities at a ®nite set of points. Using this theory we show that the generalized arithmetic selfintersection number of the line bundle of modular forms equipped with its canonical metric equals z Q À1 2 Á z H Q À1 up to a trivial factor; here, z Q s denotes the Riemann zeta function.Brought to you by | University of Arizona Authenticated Download Date | 6/10/15 7:26 AM This article contains the main results of the ®rst part of the authors thesis directed by J. Kramer at Humboldt University and supported by the Graduiertenkolleg. The author would like to thank J. Kramer for fruitful conversations; he also pro®ted from discussions with J.-B. Bost, B. Edixhoven and D. Zagier.1. Review intersection theory on arithmetic surfaces 1.1. Preliminaries. In this article an arithmetic surface XaSpec Z is an integral, 2dimensional, regular scheme de®ned over Spec Z together with a¯at, projective morphism p: X 3 Spec ZX As we consider in this article only arithmetic surfaces XaSpec Z, we will sometimes denote them by X to ease notation.Let now D 1 Y D 2 r X be two e¨ective divisors having no horizontal component in common, i.e., D 1Y hor D 2Y hor X y j, then the intersection number at the ®nite places D 1 Y D 2 fin of D 1 and D 2 is de®ned by Ku È hn, Generalized arithmetic intersection numbers 210 Brought to you by | University of Arizona Authenticated Download Date | 6/10/15 7:26 AM Brought to you by | University of Arizona Authenticated Download Date | 6/10/15 7:26 AM