Generalized arithmetic intersection numbers

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

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

“…Applying Theorem 1.14 again to the remaining integral, we obtain 1 2πi (3.12) here the vanishing of the boundary terms follows from the compatibility of this star product and the formulas for the generalized arithmetic intersection number at the infinite places in [Kü2]. Putting together all the terms yields the claim.…”

“…Notice that on X 0 (p) we have div(ϕ * F ) ∩ div(ϕ * G) = ∅. Because of (3.24) the left hand side of (3.25) equals the negative of the formula for the generalized arithmetic intersection number ϕ * F, ϕ * G ∞ at the infinite place given in Lemma 3.9 of [Kü2]. Hence we have…”

“…In other words this star product is independent of the choice we made in the desingularization and only depends on the Baily-Borel compactification X(Γ). ii) For other choices of T (m 2 ) the contribution of (3.12) has to be calculated by different formulas, which then essentially contain the generalized arithmetic intersection number at the infinite places given in [Kü2].…”

“…[Ge] Chapter V.1 and [HZ]). For basic facts on integral models of X 0 (p), the line bundle of modular forms on it, and the normalization of the corresponding Petersson metric, we refer to [Kü2]. We now describe how T (p) can be parametrized.…”

“…If p is a prime that splits in K, then T (p) is birational to the modular curve X 0 (p). We determine the star products on these T (p) using the results of [Kü2] in Theorem 3.13.…”