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Discriminants and Arakelov Euler characteristics. (English) Zbl 1042.11040

Bennett, M. A. (ed.) et al., Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters (ISBN 1-56881-126-8/hbk). 229-255 (2002).
This is an excellent article summarizing two different generalizations, due to the authors, of discriminants to arithmetic schemes with tame Galois action, namely those defined by means of Arakelov theory and coherent duality, respectively. The authors also discuss the connection of these discriminants to the conductors and \(\varepsilon\)-factors in the functional equation of the Artin-Hasse-Weil \(L\)-functions. As a consequence, a proof of a conjecture of S. Bloch concerning the conductor of an arithmetic scheme under a certain hypothesis on the special fibres is obtained (K. Kato and T. Saito obtained a more general solution to Bloch’s conjecture recently). Finally, an example involving integral models of elliptic curves is presented.
For the entire collection see [Zbl 1002.00005].

MSC:

11G35 Varieties over global fields
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