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.

Classification : 14G40, 11F72
Keywords: arithmetic Riemann-Roch theorem, pointed stable curves, hyperbolic metric, Selberg zeta function
Mot clés : Riemann-Roch arithmétique, courbes stables pointées, métrique hyperbolique, fonction zêta de Selberg

[1] E. Arbarello & M. Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Publ. Math. I.H.É.S. 88 (1998), 97-127. | Numdam | MR | Zbl

[2] M. Artin, A. Grothendieck & J.-L. Verdier, Séminaire de Géométrie Algébrique du Bois Marie 1963/64, tome 3, Lect. Notes Math. 305 (1973). | Zbl

[3] E. W. Barnes, The theory of the $G$-function, Q. J. Math. 31 (1900), 264-314. | JFM

[4] J.-M. Bismut & J.-B. Bost, Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math. 165 (1990), 1-103. | MR | Zbl

[5] J.-B. Bost, Intersection theory on arithmetic surfaces and $L_1^2$ metrics, letter dated March, 1998.

[6] M. Burger, Small eigenvalues of Riemann surfaces and graphs, Math. Z. 205 (1990), 395-420. | MR | Zbl

[7] J. I. Burgos Gil, J. Kramer & U. Kühn, Arithmetic characteristic classes of automorphic vector bundles, Doc. Math. 10 (2005), 619-716. | MR | Zbl

[8] J. I. Burgos Gil, J. Kramer & U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), 1-172. | MR | Zbl

[9] H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. 19 (1986), 409-468. | Numdam | MR | Zbl

[10] H. Darmon, F. Diamond & R. Taylor, Fermat's last theorem, in Elliptic curves, modular forms & Fermat's last theorem (Hong Kong, 1993), Int. Press, Cambridge, MA, 1997. | MR | Zbl

[11] P. Deligne, Formes modulaires et représentations $\ell$-adiques, Sém. Bourbaki, exp. no 355, Lect. Notes in Math. 179 (1969), 139-172. | Numdam | MR | Zbl

[12] P. Deligne, Le déterminant de la cohomologie, in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., 1987, 93-177. | MR | Zbl

[13] P. Deligne & D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. I.H.É.S. 36 (1969), 75-109. | Numdam | MR | Zbl

[14] P. Deligne & M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. 349, Springer, 1973, 143-316. | MR | Zbl

[15] E. D'Hoker & D. H. Phong, On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), 537-545. | MR | Zbl

[16] T. Ebel, Equivariant analytic torsion on hyperbolic Riemann surfaces and the arithmetic Lefschetz trace of an Atkin-Lehner involution on a compact Shimura curve, Thèse, Heinrich-Heine Universität, Düsseldorf, 2006.

[17] B. Edixhoven, On Néron models, divisors and modular curves, J. Ramanujan Math. Soc. 13 (1998), 157-194. | MR | Zbl

[18] R. Elkik, Fibrés d'intersections et intégrales de classes de Chern, Ann. Sci. École Norm. Sup. 22 (1989), 195-226. | Numdam | MR | Zbl

[19] G. Freixas i Montplet, Généralisations de la théorie de l'intersection arithmétique, Thèse, 2007, Université d'Orsay.

[20] G. Freixas i Montplet, Heights and metrics with logarithmic singularities, J. reine angew. Math. 627 (2009), 97-153. | MR | Zbl

[21] D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (1986), 523-540. | MR | Zbl

[22] H. Gillet & C. Soulé, Arithmetic intersection theory, Publ. Math. I.H.É.S. 72 (1990), 93-174. | Numdam | MR | Zbl

[23] H. Gillet & C. Soulé, Characteristic classes for algebraic vector bundles with Hermitian metric. I and II, Ann. of Math. 131 (1990), 163-203 and 205-238. | MR | Zbl

[24] H. Gillet & C. Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473-543. | MR | Zbl

[25] T. Hahn, Thèse, Humboldt-Universität zu Berlin, in preparation.

[26] Y. Hashimoto, Arithmetic expressions of Selberg's zeta functions for congruence subgroups, J. Number Theory 122 (2007), 324-335. | MR | Zbl

[27] D. A. Hejhal, The Selberg trace formula for ${\mathrm{PSL}}_2\left(ℝ\right)$, vol. I, II, Lect. Notes in Math. 548 (1976), 1001 (1983). | MR | Zbl

[28] H. Hida, Congruence of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), 225-261. | MR | Zbl

[29] H. Iwaniec, Spectral methods of automorphic forms, second éd., Graduate Studies in Mathematics 53, Amer. Math. Soc., 2002. | MR | Zbl

[30] L. Ji, The asymptotic behavior of Green's functions for degenerating hyperbolic surfaces, Math. Z. 212 (1993), 375-394. | MR | Zbl

[31] J. Jorgenson & R. Lundelius, Continuity of relative hyperbolic spectral theory through metric degeneration, Duke Math. J. 84 (1996), 47-81. | MR | Zbl

[32] J. Jorgenson & R. Lundelius, A regularized heat trace for hyperbolic Riemann surfaces of finite volume, Comment. Math. Helv. 72 (1997), 636-659. | MR | Zbl

[33] N. M. Katz & B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, 1985. | MR | Zbl

[34] S. Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574. | MR | Zbl

[35] F. F. Knudsen, The projectivity of the moduli space of stable curves. II, III, Math. Scand. 52 (1983), 161-212. | MR | Zbl

[36] K. Köhler & D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. I. Statement and proof, Invent. Math. 145 (2001), 333-396. | MR | Zbl

[37] U. Kühn, Generalized arithmetic intersection numbers, J. reine angew. Math. 534 (2001), 209-236. | MR | Zbl

[38] G. Laumon & L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 39, Springer, 2000. | MR | Zbl

[39] H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), 623-635. | MR | Zbl

[40] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. I.H.É.S. 47 (1977), 33-186. | Numdam | MR | Zbl

[41] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques, Invent. Math. 98 (1989), 491-498. | MR | Zbl

[42] D. Mumford, Stability of projective varieties, Enseignement Math. 23 (1977), 39-110. | MR | Zbl

[43] D. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, 1983, 271-328. | MR | Zbl

[44] D. B. Ray & I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. 98 (1973), 154-177. | MR | Zbl

[45] D. E. Rohrlich, A modular version of Jensen's formula, Math. Proc. Cambridge Philos. Soc. 95 (1984), 15-20. | MR | Zbl

[46] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229-247. | MR | Zbl

[47] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120. | MR | Zbl

[48] R. Schoen, S. A. Wolpert & S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface, in Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., 1980, 279-285. | MR | Zbl

[49] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430. | MR | Zbl

[50] M. Schulze, On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry, J. Funct. Anal. 236 (2006), 120-160. | MR | Zbl

[51] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. | MR | Zbl

[52] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), 79-98. | MR | Zbl

[53] C. Soulé, Régulateurs, Séminaire Bourbaki, vol. 1984/85, exp. no 644, Astérisque 133-134 (1986), 237-253. | Numdam | Zbl

[54] J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), 219-240. | MR | Zbl

[55] L. A. Takhtajan & P. G. Zograf, The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces, J. Geom. Phys. 5 (1988), 551-570. | MR | Zbl

[56] L. A. Takhtajan & P. G. Zograf, A local index theorem for families of $\overline{\partial }$-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces, Comm. Math. Phys. 137 (1991), 399-426. | MR | Zbl

[57] E. Ullmo, Hauteur de Faltings de quotients de $J_0\left(N\right)$, discriminants d’algèbres de Hecke et congruences entre formes modulaires, Amer. J. Math. 122 (2000), 83-115. | MR | Zbl

[58] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Soc. Math. France, 2002. | MR | Zbl

[59] A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465. | MR | Zbl

[60] L. Weng, $\Omega$-admissible theory. II. Deligne pairings over moduli spaces of punctured Riemann surfaces, Math. Ann. 320 (2001), 239-283. | MR | Zbl

[61] S. A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), 283-315. | MR | Zbl

[62] S. A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), 417-472. | MR | Zbl

[63] S. A. Wolpert, Cusps and the family hyperbolic metric, Duke Math. J. 138 (2007), 423-443. | MR | Zbl