×

La formule de Noether pour les surfaces arithmétiques. (The Noether formula for arithmetic surfaces). (French) Zbl 0727.14014

Let \(\bar {\mathcal M}_ g\) be the algebraic stack over Spec(\({\mathbb{Z}})\) of stable curves of genus \(g,\) \({\mathcal M}_ g\) the open part corresponding to smooth curves, \(\Delta\) the normally crossing divisor supported by \(\bar {\mathcal M}_ g\setminus {\mathcal M}_ g\), p: \({\mathcal C}\to {\mathcal M}_ g\) the universal curve and \(K=\omega_{{\mathcal C}/\bar {\mathcal M}_ g}\) the relative canonical sheaf. Using Deligne’s pairing \(<, >\) the author presents Mumford’s isomorphism [D. Mumford, Enseign. Math., II. Sér. 23, 39-100 (1977; Zbl 0363.14003)], uniquely up to sign, in the form \((\det (p_*K))^{\otimes 12}\overset \sim \rightarrow <K,K>\otimes {\mathcal O}_{\bar {\mathcal M}_ g}(\Delta)\) (theorem 2.1) and proves that the natural hermitian norms on both invertible sheaves are connected by multiplication with \((2\pi)^{-4g}\) \(e^{\delta}\) along this isomorphism (theorem 2.2), where \(\delta\) is the real function on \({\mathcal M}_ g({\mathbb{C}})\) defined by G. Faltings in Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005). - The proof applies the functorial theory of relative curves (X/S,\(\ell)\) with theta characteristics \(\ell\) \((2\ell=\) class of \(\Omega^ 1_{X/S})\), in analogy with the paper by A. A. Bejlinson and Yu. I. Manin [Commun. Math. Phys. 107, 359-376 (1986; Zbl 0604.14016)].
Clearly, the basic theorem implies the Mumford isomorphism and Faltings’ Noether formula for arithmetic surfaces X/B: \[ 12\cdot \deg (\det (Rp_*\omega))=(\omega \cdot \omega)+\sum_{b}\delta_ b(X)\cdot \log (N(b)) +\sum_{\sigma}\delta_{\sigma}(X) -4g[L:{\mathbb{Q}}]\log (2\pi), \] where \(B=Spec(R)\), R the ring of integers of a number field L, \(\omega =\omega_{X/B}\), N(b) the absolute norm of \(b\in Spec(R)\), ( \(\cdot)\) is Arakelov’s intersection pairing, and the \(\sigma\) denote field embeddings of L into \({\mathbb{C}}\).

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14H10 Families, moduli of curves (algebraic)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] [A] Arakelov, S.Ju.: Intersection theory of divisors on an arithmetic surface. Math. USSR. Izv.8, 1167-1180 (1974) · Zbl 0355.14002
[2] [B-M] Beilinson, A.A., Manin, Yu.I.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys.107, 359-376 (1986) · Zbl 0604.14016
[3] [D] Deligne, P.: Le déterminant de la cohomologie. Contemp. Math.67, 93-177 (1987)
[4] [D-M] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Pub. Math. Inst. Hautes Etud. Sci., vol. 36 · Zbl 0233.14008
[5] [D-R] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Deligne, P., Kuijk, W. (eds.) Modular functions of one variable II. Proceedings Antwerp 1972 (Lect. Notes Math., Vol. 349). Berlin Heidelberg New York: Springer
[6] [F] Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387-424 (1984) · Zbl 0559.14005
[7] [M] Mumford, D.: Stability of projective varieties. Enseign. math.23, 39-100 (1977) · Zbl 0363.14003
[8] [MB1] Moret-Bailly, L.: Pinceaux de variétés abéliennes. Astérisque, vol. 129
[9] [MB2] Moret-Bailly, L.: Sur l’équation fonctionnelle de la fonction thêta de Riemann (preprint)
[10] [Sai] Saito, T.: Conductor, discriminant and the Noether formula of arithmetic surfaces. Duke Math. J.57, 151-173 (1988) · Zbl 0657.14017
[11] [Sat] Satake, I.: On the compactification of the Siegel space. J. Indian Math. Soc.20, 259-281 (1956) · Zbl 0072.30002
[12] [SGA4] Artin, M., Grothendieck, A., Verdier, J.-L.: Séminaire de Géométrie Algébrique du Bois Marie 1963/64, tome 3 (Lect. Notes Math., Vol. 305). Berlin Heidelberg New York: Springer
[13] [SPA] Szpiro, L.: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell. Astérisque, vol. 127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.