Bounds on Faltings’s delta function through covers

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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      PAGES = {148--211},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {58G25 (11F72 47F05 81E30)},
      MRNUMBER = {90d:58159},
      DOI = {10.1016/0022-1236(88)90070-5},
      ZBLNUMBER = {0653.53022},
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      }
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      PAGES = {113--120},
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      INVSERIES = {S{é}minaire Bourbaki, Vol. 1988/89},
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      VOLUME = {177-178},
      YEAR = {1989},
      PAGES = {Exp. No. 713, 327--343},
      ISSN = {0303-1179},
      MRCLASS = {14G40 (11G30 14C17 14C40)},
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      }
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      1982},
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    @incollection{kn:Zograf, MRKEY = {661463},
      AUTHOR = {Zograf, P. G.},
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      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)},
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      number = {122},
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      MRREVIEWER = {P. G{ü}nther},
      }

Authors

Jay Jorgenson

Department of Mathematics
City College of New York
Convent Avenue at 138th Street
New York, NY 10031
United States

Jürg Kramer

Institut für Mathematik
Humboldt-Universität zu Berlin
Unter den Linden 6
D-10099 Berlin
Germany