An arithmetic Riemann-Roch theorem for pointed stable curves  [ Un théorème de Riemann-Roch arithmétique pour les courbes stables pointées ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 2, p. 335-369
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.
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.
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
@article{ASENS_2009_4_42_2_335_0,
     author = {Freixas Montplet, G\'erard},
     title = {An arithmetic Riemann-Roch theorem for pointed stable curves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {2},
     year = {2009},
     pages = {335-369},
     doi = {10.24033/asens.2098},
     zbl = {1183.14038},
     mrnumber = {2518081},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_2_335_0}
}
Freixas Montplet, Gérard. An arithmetic Riemann-Roch theorem for pointed stable curves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 2, pp. 335-369. doi : 10.24033/asens.2098. https://www.numdam.org/item/ASENS_2009_4_42_2_335_0/

[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 1733327 | Zbl 0991.14012

[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 0269.14010

[3] E. W. Barnes, The theory of the G-function, Q. J. Math. 31 (1900), 264-314. | JFM 30.0389.02

[4] J.-M. Bismut & J.-B. Bost, Fibrés déterminants, métriques de Quillen et dégénérescence des courbes, Acta Math. 165 (1990), 1-103. | MR 1064578 | Zbl 0709.32019

[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 1082864 | Zbl 0729.58050

[7] J. I. Burgos Gil, J. Kramer & U. Kühn, Arithmetic characteristic classes of automorphic vector bundles, Doc. Math. 10 (2005), 619-716. | MR 2218402 | Zbl 1080.14028

[8] J. I. Burgos Gil, J. Kramer & U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), 1-172. | MR 2285241 | Zbl 1115.14013

[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 870690 | Zbl 0616.10025

[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 1605752 | Zbl 0877.11035

[11] P. Deligne, Formes modulaires et représentations -adiques, Sém. Bourbaki, exp. no 355, Lect. Notes in Math. 179 (1969), 139-172. | Numdam | MR 3077124 | Zbl 0206.49901

[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 902592 | Zbl 0629.14008

[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 262240 | Zbl 0181.48803

[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 337993 | Zbl 0281.14010

[15] E. D'Hoker & D. H. Phong, On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), 537-545. | MR 841668 | Zbl 0599.30073

[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 1666374 | Zbl 0931.11021

[18] R. Elkik, Fibrés d'intersections et intégrales de classes de Chern, Ann. Sci. École Norm. Sup. 22 (1989), 195-226. | Numdam | MR 1005159 | Zbl 0701.14003

[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 2494930 | Zbl 1195.14033

[21] D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (1986), 523-540. | MR 837526 | Zbl 0621.53035

[22] H. Gillet & C. Soulé, Arithmetic intersection theory, Publ. Math. I.H.É.S. 72 (1990), 93-174. | Numdam | MR 1087394 | Zbl 0741.14012

[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 1038362 | Zbl 0715.14006

[24] H. Gillet & C. Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), 473-543. | MR 1189489 | Zbl 0777.14008

[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 2292258 | Zbl 1163.11060

[27] D. A. Hejhal, The Selberg trace formula for PSL 2 (), vol. I, II, Lect. Notes in Math. 548 (1976), 1001 (1983). | MR 711197 | Zbl 0543.10020

[28] H. Hida, Congruence of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), 225-261. | MR 610538 | Zbl 0459.10018

[29] H. Iwaniec, Spectral methods of automorphic forms, second éd., Graduate Studies in Mathematics 53, Amer. Math. Soc., 2002. | MR 1942691 | Zbl 1006.11024

[30] L. Ji, The asymptotic behavior of Green's functions for degenerating hyperbolic surfaces, Math. Z. 212 (1993), 375-394. | MR 1207299 | Zbl 0792.53040

[31] J. Jorgenson & R. Lundelius, Continuity of relative hyperbolic spectral theory through metric degeneration, Duke Math. J. 84 (1996), 47-81. | MR 1394748 | Zbl 0872.58062

[32] J. Jorgenson & R. Lundelius, A regularized heat trace for hyperbolic Riemann surfaces of finite volume, Comment. Math. Helv. 72 (1997), 636-659. | MR 1600164 | Zbl 0902.58040

[33] N. M. Katz & B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, 1985. | MR 772569 | Zbl 0576.14026

[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 1034665 | Zbl 0768.14002

[35] F. F. Knudsen, The projectivity of the moduli space of stable curves. II, III, Math. Scand. 52 (1983), 161-212. | MR 702953 | Zbl 0544.14020

[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 1872550 | Zbl 0999.14002

[37] U. Kühn, Generalized arithmetic intersection numbers, J. reine angew. Math. 534 (2001), 209-236. | MR 1831639 | Zbl 1084.14028

[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 1771927 | Zbl 0945.14005

[39] H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), 623-635. | MR 417456 | Zbl 0358.32017

[40] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. I.H.É.S. 47 (1977), 33-186. | Numdam | MR 488287 | Zbl 0394.14008

[41] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques, Invent. Math. 98 (1989), 491-498. | MR 1022303 | Zbl 0727.14014

[42] D. Mumford, Stability of projective varieties, Enseignement Math. 23 (1977), 39-110. | MR 450272 | Zbl 0363.14003

[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 717614 | Zbl 0554.14008

[44] D. B. Ray & I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. 98 (1973), 154-177. | MR 383463 | Zbl 0267.32014

[45] D. E. Rohrlich, A modular version of Jensen's formula, Math. Proc. Cambridge Philos. Soc. 95 (1984), 15-20. | MR 727075 | Zbl 0538.10023

[46] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229-247. | MR 675187 | Zbl 0499.10021

[47] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120. | MR 885573 | Zbl 0618.10023

[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 573440 | Zbl 0446.58018

[49] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430. | MR 1047142 | Zbl 0760.14002

[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 2227131 | Zbl 1094.58012

[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 314766 | Zbl 0221.10029

[52] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), 79-98. | MR 382176 | Zbl 0311.10029

[53] C. Soulé, Régulateurs, Séminaire Bourbaki, vol. 1984/85, exp. no 644, Astérisque 133-134 (1986), 237-253. | Numdam | Zbl 0617.14008

[54] J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), 219-240. | MR 564472 | Zbl 0433.10015

[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 1075722 | Zbl 0739.30032

[56] L. A. Takhtajan & P. G. Zograf, A local index theorem for families of ¯-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces, Comm. Math. Phys. 137 (1991), 399-426. | MR 1101693 | Zbl 0725.58043

[57] E. Ullmo, Hauteur de Faltings de quotients de J 0 (N), discriminants d’algèbres de Hecke et congruences entre formes modulaires, Amer. J. Math. 122 (2000), 83-115. | MR 1737258 | Zbl 0991.11033

[58] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Soc. Math. France, 2002. | MR 1988456 | Zbl 1032.14001

[59] A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465. | MR 891947 | Zbl 0631.10025

[60] L. Weng, Ω-admissible theory. II. Deligne pairings over moduli spaces of punctured Riemann surfaces, Math. Ann. 320 (2001), 239-283. | MR 1839763 | Zbl 1036.14013

[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 905169 | Zbl 0629.58029

[62] S. A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), 417-472. | MR 1037410 | Zbl 0698.53002

[63] S. A. Wolpert, Cusps and the family hyperbolic metric, Duke Math. J. 138 (2007), 423-443. | MR 2322683 | Zbl 1144.14029