An arithmetic Riemann-Roch theorem for pointed stable curves

Gérard Freixas Montplet

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 2, page 335-369
  • ISSN: 0012-9593

Abstract

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Let ( 𝒪 , Σ , F ∞ ) be an arithmetic ring of Krull dimension at most 1, 𝒮 = Spec 𝒪 and ( π : 𝒳 → 𝒮 ; 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 ) , p ≥ 11 prime, with sections given by the cusps. We show Z ' ( Y ( Γ ) , 1 ) ∼ e a π b Γ 2 ( 1 / 2 ) c L ( 0 , ℳ Γ ) , with p ≡ 11 m o d 12 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, ℳ Γ is a suitable Chow motive and ∼ means equality up to algebraic unit.

How to cite

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Freixas Montplet, Gérard. "An arithmetic Riemann-Roch theorem for pointed stable curves." Annales scientifiques de l'École Normale Supérieure 42.2 (2009): 335-369. <http://eudml.org/doc/272144>.

@article{FreixasMontplet2009,
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.},
author = {Freixas Montplet, Gérard},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {arithmetic Riemann-Roch theorem; pointed stable curves; hyperbolic metric; Selberg zeta function},
language = {eng},
number = {2},
pages = {335-369},
publisher = {Société mathématique de France},
title = {An arithmetic Riemann-Roch theorem for pointed stable curves},
url = {http://eudml.org/doc/272144},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Freixas Montplet, Gérard
TI - An arithmetic Riemann-Roch theorem for pointed stable curves
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 2
SP - 335
EP - 369
AB - 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.
LA - eng
KW - arithmetic Riemann-Roch theorem; pointed stable curves; hyperbolic metric; Selberg zeta function
UR - http://eudml.org/doc/272144
ER -

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