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Examples and Results

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Conjectures in Arithmetic Algebraic Geometry

Part of the book series: Aspects of Mathematics ((ASMA,volume 18))

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Abstract

In this final chapter we discuss in some detail several results related to the conjectures of the foregoing chapters. The first result concerns important work of B. Gross and D. Zagier on the Birch & Swinnerton-Dyer Conjectures. Next, an overview of Deligne’s Conjecture on the L-function of an algebraic Hecke character is given. This conjecture is now a theorem, due to work of D. Blasius, G. Harder and N. Schappacher. The third and fourth sections treat Beilinson’s results on regulators for Artin motives and modular curves, respectively. In this last situation a (possibly) general phenomenon occurs: only part of motivic co-homology is useful. This phenomenon was already encountered in the discussion of Ramakrishnan’s result on the regulator map for Hilbert modular surfaces. In the last section a class of varieties is introduced for which the Hodge and Tate Conjectures are true. This result is due to U. Jannsen.

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References and Suggestions for Further Reading

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Authors and Affiliations

  1. KMA, NL-4800 RG, Breda, The Netherlands

    Wilfred W. J. Hulsbergen

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  1. Wilfred W. J. Hulsbergen
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© 1994 Springer Fachmedien Wiesbaden

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Hulsbergen, W.W.J. (1994). Examples and Results. In: Conjectures in Arithmetic Algebraic Geometry. Aspects of Mathematics, vol 18. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-09505-7_11

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  • DOI: https://doi.org/10.1007/978-3-663-09505-7_11

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-663-09507-1

  • Online ISBN: 978-3-663-09505-7

  • eBook Packages: Springer Book Archive

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