## 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

### Citations:

Zbl 0363.14003; Zbl 0559.14005; Zbl 0604.14016
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### References:

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