Bounds on Faltings’s delta function through covers

Jay Jorgenson; Jürg Kramer

doi:10.4007/annals.2009.170.1

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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@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.

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@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}, }

Bounds on Faltings’s delta function through covers

Abstract

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