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Conjectures in Arithmetic Algebraic Geometry pp 137–143Cite as

Arithmetic intersections and Beilinson’s third conjecture

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

Abstract

This chapter treats the case of odd i and m = EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaadaWcaaqaaiaadMgacqGHRaWkcaaIXaaa % baGaaGOmaaaaaaa!4277!</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \frac{{i + 1}}{2} $$. The L-functions considered in the sequel will be those defined by the cohomology space Hi. Beilinson’s third conjecture regards this situation for smooth, projective varieties over ℚ, and reduces to a weakened form of the Birch & Swinnerton-Dyer Conjectures in the case of an elliptic curve or an abelian variety over ℚ. The elliptic regulator is generalized to become the determinant of an arithmetic intersection index on arithmetic varieties on Spec(ℤ), thus enlarging Arakelov’s construction of the Néron-Tate height pairing. This generalized height pairing was constructed by Beilinson and, independently, by Gillet and Soulé. In [Bl4] Bloch defines another height pairing for algebraic cycles.

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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). Arithmetic intersections and Beilinson’s third conjecture. 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_8

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  • DOI: https://doi.org/10.1007/978-3-663-09505-7_8

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-663-09507-1

  • Online ISBN: 978-3-663-09505-7

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