Soit un entier naturel impair sans facteur carré ayant au moins deux diviseurs relativement premiers et supérieurs ou égaux à . Le théorème principal de cet article est une formule asymptotique exclusivement en termes de pour l’auto-intersection arithmétique du dualisant relatif des courbes modulaires . Nous en déduisons une formule asymptotique pour la hauteur stable de Faltings de la Jacobienne de ainsi qu’une version effective de la conjecture de Bogomolov pour pour suffisamment grand.
Let be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than . Our main theorem is an asymptotic formula solely in terms of for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves . From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian of , and, for sufficiently large , an effective version of Bogomolov’s conjecture for .
@article{JTNB_2014__26_1_111_0, author = {Mayer, Hartwig}, title = {Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {111--161}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {1}, year = {2014}, doi = {10.5802/jtnb.862}, zbl = {06304184}, mrnumber = {3232770}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.862/} }
TY - JOUR AU - Mayer, Hartwig TI - Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$ JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 111 EP - 161 VL - 26 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.862/ DO - 10.5802/jtnb.862 LA - en ID - JTNB_2014__26_1_111_0 ER -
%0 Journal Article %A Mayer, Hartwig %T Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$ %J Journal de théorie des nombres de Bordeaux %D 2014 %P 111-161 %V 26 %N 1 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.862/ %R 10.5802/jtnb.862 %G en %F JTNB_2014__26_1_111_0
Mayer, Hartwig. Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 111-161. doi : 10.5802/jtnb.862. http://archive.numdam.org/articles/10.5802/jtnb.862/
[1] A. Abbes and E. Ullmo, Auto-intersection du dualisant relatif des courbes modulaires . J. Reine Angew. Math. 484 (1997), 1–70. | MR | Zbl
[2] S. J. Arakelov, Intersection theory of divisors on an arithmetic surface. Math. USSR Izvestija 8 (1974), 1167–1180. | MR | Zbl
[3] A. O. L. Atkin and W.-C. W. Li, Twists with Newforms and Pseudo-Eigenvalues of -Operators. Invent. Math. 48 (1978), 221–243. | MR | Zbl
[4] J.-B. Bost, J.-F. Mestre and L. Moret-Bailly, Sur le calcul explicite des “classes de Chern” des surfaces arithmétiques de genre . In: Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988), Astérisque 183 (1990), 69–105. | Numdam | MR | Zbl
[5] C. Curilla and U. Kühn, On the arithmetic self-intersection numbers of the dualizing sheaf for Fermat curves of prime exponent. arXiv:0906.3891v1, 2009.
[6] F. Diamond and J. Shurman, A first course in modular forms. Graduate Texts in Mathematics 228, Springer-Verlag, 2005. | MR | Zbl
[7] V. G. Drinfeld, Two theorems on modular curves. Funct. Anal. Appl. 7 (1973), 155–156. | MR | Zbl
[8] B. Edixhoven and J.-M. Couveignes et al., Computational aspects of modular forms and Galois representations. Annals of Mathematics Studies 176, Princeton University Press, 2011. | MR | Zbl
[9] G. Faltings, Calculus on arithmetic surfaces. Ann. of Math. 119 (1984), 387–424. | MR | Zbl
[10] S. D. Gupta, The Rankin-Selberg Method on Congruence Subgroups. Illinois J. Math. 44 (2000), 95–103. | MR | Zbl
[11] D. A. Hejhal The Selberg trace formula for . Vol. 1. Lecture Notes in Mathematics Vol. 548, Springer-Verlag, Berlin, 1976. | MR | Zbl
[12] D. A. Hejhal The Selberg trace formula for . Vol. 2. Lecture Notes in Mathematics Vol. 1001, Springer-Verlag, Berlin, 1983. | MR | Zbl
[13] H. Iwaniec, Spectral methods of automorphic forms. Graduate Studies in Mathematics 53, American Mathematical Society, 2002. | MR | Zbl
[14] H. Jacquet and D. Zagier, Eisenstein series and the Selberg trace formula. II. Trans. Amer. Math. Soc. 300 (1987), 1–48. | MR | Zbl
[15] J. Jorgenson and J. Kramer, Bounds for special values of Selberg zeta functions of Riemann surfaces. J. Reine Angew. Math. 541 (2001), 1–28. | MR | Zbl
[16] J. Jorgenson and J. Kramer, Bounds on canonical Green’s functions. Compos. Math. 142 (2006), 679–700. | MR | Zbl
[17] J. Jorgenson and J. Kramer, Bounds on Faltings’s delta function through covers. Ann. of Math. 1 (2009), 1–43. | MR | Zbl
[18] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves. Princeton University Press, 1985. | MR | Zbl
[19] C. Keil, Die Streumatrix für Untergruppen der Modulgruppe. PhD thesis, Universität Frankfurt am Main, 2006.
[20] E. Landau, Neuer Beweis eines analytischen Satzes des Herrn de la Vallée Poussin. Math. Ann. 56 (1903), 645–670. | MR
[21] H. Mayer, Self-Intersection of the Relative Dualizing Sheaf of Modular Curves . PhD thesis, Humboldt-Universität zu Berlin, 2012.
[22] H. P. McKean, Selberg’s trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math. 25 (1972), 225–246. | MR
[23] P. Michel and E. Ullmo, Points de petite hauteur sur les courbes modulaires . Invent. Math. 131 (1998), 645–674. | MR | Zbl
[24] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques. Invent. Math. 98 (1989), 491–498. | MR | Zbl
[25] A. P. Ogg, Rational Points on certain Elliptic modular curves. In: Analytic number theory (Proc. Sympos. Pure Math., Vol XXIV), Amer. Math. Soc., Providence, RI (1973), 221–231. | MR | Zbl
[26] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene I. Math. Ann. 167 (1966), 292–337. | MR | Zbl
[27] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene II. Math. Ann. 168 (1967), 261–324. | MR | Zbl
[28] C. Soulé, Géométrie d’Arakelov des surfaces arithmétiques. In: Séminaire Bourbaki, Vol. 1988/89, Astérisque 177-178 (1989), 327–343. | Numdam | MR | Zbl
[29] L. Szpiro, Sur les propriétés numériques du dualisant relatif d’une surface arithmétique. In: The Grothendieck Festschrift, Vol. III, Birkhäuser Boston (1990), 229–246. | MR | Zbl
[30] D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular functions of one variable VI, Lect. Notes Math. 627, Springer-Verlag (1977), 105–169. | MR | Zbl
[31] D. Zagier, Eisenstein series and the Selberg trace formula. I. In: Automorphic forms, representation theory and arithmetic (Bombay, 1979). Tata Inst. Fund. Res. Studies in Math. 10 (1981), 303–355. | MR | Zbl
[32] D. Zagier, Zetafunktionen und quadratische Zahlkörper. Springer-Verlag, Berlin, 1981. | MR
[33] S.-W. Zhang, Admissible pairing on a curve. Invent. Math. 112 (1993), 171–193. | MR | Zbl
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