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 \({\mathbb{C}}\).


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
Full Text: DOI EuDML


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