An arithmetic Riemann-Roch theorem for pointed stable curves
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, pp. 335-369.

Let (𝒪,Σ,F ) be an arithmetic ring of Krull dimension at most 1, 𝒮= Spec 𝒪 and (π:𝒳𝒮;σ 1 ,...,σ n ) an n-pointed stable curve of genus g. Write 𝒰=𝒳 j σ j (𝒮). The invertible sheaf ω 𝒳/𝒮 (σ 1 ++σ n ) inherits a hermitian structure · hyp from the dual of the hyperbolic metric on the Riemann surface 𝒰 . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω 𝒳/𝒮 (σ 1 +...+σ n ) hyp . The theorem is applied to modular curves X(Γ), Γ=Γ 0 (p) or Γ 1 (p), p11 prime, with sections given by the cusps. We show Z ' (Y(Γ),1)e a π b Γ 2 (1/2) c L(0, Γ ), with p11mod12 when Γ=Γ 0 (p). Here Z(Y(Γ),s) is the Selberg zeta function of the open modular curve Y(Γ), a,b,c are rational numbers, Γ is a suitable Chow motive and means equality up to algebraic unit.

Soient (𝒪,Σ,F ) un anneau arithmétique de dimension de Krull au plus 1, 𝒮= Spec 𝒪 et (π:𝒳𝒮;σ 1 ,...,σ n ) une courbe stable n-pointée de genre g. Posons 𝒰=𝒳 j σ j (𝒮). Le faisceau inversible ω 𝒳/𝒮 (σ 1 ++σ n ) hérite une structure hermitienne · hyp du dual de la métrique hyperbolique sur la surface de Riemann 𝒰 . Dans cet article nous prouvons un théorème de Riemann-Roch arithmétique qui calcule l’auto-intersection arithmétique de ω 𝒳/𝒮 (σ 1 ++σ n ) hyp . Le théorème est appliqué aux courbes modulaires X(Γ), Γ=Γ 0 (p) ou Γ 1 (p), p11 premier, prenant les cusps comme sections. Nous montrons Z ' (Y(Γ),1)e a π b Γ 2 (1/2) c L(0, Γ ), avec p11mod12 lorsque Γ=Γ 0 (p). Ici Z(Y(Γ),s) est la fonction zêta de Selberg de la courbe modulaire ouverte Y(Γ), a,b,c sont des nombres rationnels, Γ est un motif de Chow approprié et signifie égalité à unité près.

DOI: 10.24033/asens.2098
Classification: 14G40,  11F72
Keywords: arithmetic Riemann-Roch theorem, pointed stable curves, hyperbolic metric, Selberg zeta function
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     title = {An arithmetic {Riemann-Roch} theorem for pointed stable curves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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     publisher = {Soci\'et\'e math\'ematique de France},
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Freixas Montplet, Gérard. An arithmetic Riemann-Roch theorem for pointed stable curves. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, pp. 335-369. doi : 10.24033/asens.2098. http://archive.numdam.org/articles/10.24033/asens.2098/

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