An arithmetic Riemann-Roch theorem for pointed stable curves

Freixas Montplet, Gérard

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 2, pp. 335-369.

Soient $(𝒪,\Sigma ,F_{\infty })$ un anneau arithmétique de dimension de Krull au plus 1, $𝒮=\mathrm{Spec}𝒪$ et $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ une courbe stable $n$-pointée de genre $g$. Posons $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. Le faisceau inversible ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ hérite une structure hermitienne ${\parallel ·\parallel }_{\mathrm{hyp}}$ du dual de la métrique hyperbolique sur la surface de Riemann ${𝒰}_{\infty }$. Dans cet article nous prouvons un théorème de Riemann-Roch arithmétique qui calcule l’auto-intersection arithmétique de ${\omega }_{𝒳/𝒮}{({\sigma }_1+\cdots +{\sigma }_n)}_{\mathrm{hyp}}$. Le théorème est appliqué aux courbes modulaires $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ ou ${\Gamma }_1\left(p\right)$, $p\ge 11$ premier, prenant les cusps comme sections. Nous montrons $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, avec $p\equiv 11mod12$ lorsque $\Gamma ={\Gamma }_0\left(p\right)$. Ici $Z\left(Y\right(\Gamma ),s)$ est la fonction zêta de Selberg de la courbe modulaire ouverte $Y\left(\Gamma \right)$, $a,b,c$ sont des nombres rationnels, ${ℳ}_{\Gamma }$ est un motif de Chow approprié et $\sim$ signifie égalité à unité près.

Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg zeta function of the open modular curve $Y\left(\Gamma \right)$, $a,b,c$ are rational numbers, ${ℳ}_{\Gamma }$ is a suitable Chow motive and $\sim$ means equality up to algebraic unit.

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