Abstract
Let $X$ be a compact Riemann surface of genus $g_{X}\geq1$. In 1984, G. Faltings introduced a new invariant $\delta_{\operatorname{Fal}}(X)$ associated to $X$. In this paper we give explicit bounds for $\delta_{\operatorname{Fal}}(X)$ in terms of fundamental differential geometric invariants arising from $X$, when $g_{X}>1$. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves $X_{0}(N)$ in terms of the genus only. In combination with work of A. Abbes, P. Michel and E. Ullmo, this leads to an asymptotic formula for the Faltings height of the Jacobian $J_{0}(N)$ associated to $X_{0}(N)$.
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[kn:Soule] C. Soulé, "Géométrie d’Arakelov des surfaces arithmétiques," Astérisque, vol. 177-178, p. exp. no. 713, 327-343, 1989.
@article{kn:Soule, MRKEY = {1040579},
AUTHOR = {Soul{é},
Christophe},
TITLE = {Géométrie d'{A}rakelov des surfaces arithmétiques},
INVSERIES = {S{é}minaire Bourbaki, Vol. 1988/89},
JOURNAL = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {177-178},
YEAR = {1989},
PAGES = {Exp. No. 713, 327--343},
ISSN = {0303-1179},
MRCLASS = {14G40 (11G30 14C17 14C40)},
MRNUMBER = {91c:14031},
ZBLNUMBER = {0766.14015},
MRREVIEWER = {Philippe Satg{é}},
} -
[kn:Ullmo]
E. Ullmo, "Hauteur de Faltings de quotients de $J_0(N)$, discriminants d’algèbres de Hecke et congruences entre formes modulaires," Amer. J. Math., vol. 122, iss. 1, pp. 83-115, 2000.
@article{kn:Ullmo, MRKEY = {1737258},
AUTHOR = {Ullmo, Emmanuel},
TITLE = {Hauteur de {F}altings de quotients de {$J\sb 0(N)$},
discriminants d'algèbres de {H}ecke et congruences entre formes modulaires},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {122},
YEAR = {2000},
NUMBER = {1},
PAGES = {83--115},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {11G50 (11F33 11G18 14G40)},
MRNUMBER = {2000k:11080},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.1ullmo.pdf},
ZBLNUMBER = {0991.11033},
MRREVIEWER = {Carlo Gasbarri},
} -
[kn:Venkov] A. B. Venkov, "Spectral theory of automorphic functions," Trudy Mat. Inst. Steklov., vol. 153, p. 172, 1981.
@article{kn:Venkov, MRKEY = {665585},
AUTHOR = {Venkov, A. B.},
TITLE = {Spectral theory of automorphic functions},
JOURNAL = {Trudy Mat. Inst. Steklov.},
FJOURNAL = {Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova},
NOTE = {in Russian; translated in {\it Proc. Steklov Inst. Math.} {\bf 153},
1982},
VOLUME = {153},
YEAR = {1981},
PAGES = {172},
ISSN = {0371-9685},
MRCLASS = {11F72 (11F03 11M41 22E45 30F35 58G25)},
MRNUMBER = {85j:11060a},
ZBLNUMBER = {0483.10029},
MRREVIEWER = {J{ü}rgen Elstrodt},
} -
[kn:Zograf] P. G. Zograf, "Fuchsian groups and small eigenvalues of the Laplace operator," in Studies in Topology, IV, , 1982, pp. 24-29, 163.
@incollection{kn:Zograf, MRKEY = {661463},
AUTHOR = {Zograf, P. G.},
TITLE = {Fuchsian groups and small eigenvalues of the {L}aplace operator},
booktitle = {Studies in Topology, IV},
series = {Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. {$($LOMI$)$}},
FJOURNAL = {Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI)},
NOTE = {in Russian; translated in {\it Soviet Math. Dokl\/}. {\bf 26} (1982), 24--29},
number = {122},
YEAR = {1982},
PAGES = {24--29, 163},
ISSN = {0206-8540},
MRCLASS = {58G25 (10D12 30F35)},
MRNUMBER = {84a:58090},
ZBLNUMBER = {0536.10017},
MRREVIEWER = {P. G{ü}nther},
}