An arithmetic Riemann-Roch theorem for pointed stable curves

[Un théorème de Riemann-Roch arithmétique pour les courbes stables pointées]

Freixas Montplet, Gérard

Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 335-369 / Harvested from Numdam

Résumé

Soient $(\mathcal {O}, \Sigma , F_{\infty })$ un anneau arithmétique de dimension de Krull au plus 1, $\mathcal {S}=\mathrm {Spec}\mathcal {O}$ et $(\pi :\mathcal {X}\rightarrow \mathcal {S}; \sigma _{1},\ldots ,\sigma _{n})$ une courbe stable $n$-pointée de genre $g$. Posons $\mathcal {U}=\mathcal {X}\setminus \cup _{j}\sigma _{j}(\mathcal {S})$. Le faisceau inversible $\omega _{\mathcal {X}/\mathcal {S}}(\sigma _{1}+\cdots +\sigma _{n})$ hérite une structure hermitienne $\Vert \cdot \Vert _{\mathrm {hyp}}$ du dual de la métrique hyperbolique sur la surface de Riemann $\mathcal {U}_{\infty }$. Dans cet article nous prouvons un théorème de Riemann-Roch arithmétique qui calcule l’auto-intersection arithmétique de $\omega _{\mathcal {X}/\mathcal {S}}(\sigma _{1}+\cdots +\sigma _{n})_{\mathrm {hyp}}$. Le théorème est appliqué aux courbes modulaires $X(\Gamma )$, $\Gamma =\Gamma _{0}(p)$ ou $\Gamma _{1}(p)$, $p\ge 11$ premier, prenant les cusps comme sections. Nous montrons $Z^{\prime }(Y(\Gamma ),1)\sim e^{a}\pi ^{b}\Gamma _{2}(1/2)^{c}L(0,\mathcal {M}_{\Gamma })$, avec $p\equiv 11~mod \;12$ lorsque $\Gamma =\Gamma _{0}(p)$. Ici $Z(Y(\Gamma ),s)$ est la fonction zêta de Selberg de la courbe modulaire ouverte $Y(\Gamma )$, $a,b,c$ sont des nombres rationnels, $\mathcal {M}_{\Gamma }$ est un motif de Chow approprié et $\sim $ signifie égalité à unité près.

Abstract

Let $(\mathcal {O}, \Sigma , F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $\mathcal {S}=\mathrm {Spec}\mathcal {O}$ and $(\pi :\mathcal {X}\rightarrow \mathcal {S}; \sigma _{1},\ldots ,\sigma _{n})$ an $n$-pointed stable curve of genus $g$. Write $\mathcal {U}=\mathcal {X}\setminus \cup _{j}\sigma _{j}(\mathcal {S})$. The invertible sheaf $\omega _{\mathcal {X}/\mathcal {S}}(\sigma _{1}+\cdots +\sigma _{n})$ inherits a hermitian structure $\Vert \cdot \Vert _{\mathrm {hyp}}$ from the dual of the hyperbolic metric on the Riemann surface $\mathcal {U}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of $\omega _{\mathcal {X}/\mathcal {S}}(\sigma _{1}+\ldots +\sigma _{n})_{\mathrm {hyp}}$. The theorem is applied to modular curves $X(\Gamma )$, $\Gamma =\Gamma _{0}(p)$ or $\Gamma _{1}(p)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\prime }(Y(\Gamma ),1)\sim e^{a}\pi ^{b}\Gamma _{2}(1/2)^{c}L(0,\mathcal {M}_{\Gamma })$, with $p\equiv 11~mod \;12$ when $\Gamma =\Gamma _{0}(p)$. Here $Z(Y(\Gamma ),s)$ is the Selberg zeta function of the open modular curve $Y(\Gamma )$, $a,b,c$ are rational numbers, $\mathcal {M}_{\Gamma }$ is a suitable Chow motive and $\sim $ means equality up to algebraic unit.

Détail

Publié le : 2009-01-01

MR 2518081 | Zbl 1183.14038

DOI : https://doi.org/10.24033/asens.2098
Classification:  14G40,  11F72
Mots clés: Riemann-Roch arithmétique, courbes stables pointées, métrique hyperbolique, fonction zêta de Selberg

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