Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometry of Fermat adeles

Author: Alexandru Buium
Journal: Trans. Amer. Math. Soc. 357 (2005), 901-964
MSC (2000): Primary 11G05, 11G30
Published electronically: October 19, 2004
MathSciNet review: 2110426
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Abstract: If $L(a,s):=\sum_n c(n,a)n^{-s}$ is a family of ``geometric'' $L-$functions depending on a parameter $a$, then the function $(p,a)\mapsto c(p,a)$, where $p$ runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by ``adjoining a Fermat quotient operation'', then the functions $c(p,a)$ become functions in the enlarged geometry at least for $L-$functions of curves and Abelian varieties.

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Alexandru Buium
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131

Received by editor(s): August 16, 2000
Received by editor(s) in revised form: May 14, 2002
Published electronically: October 19, 2004
Additional Notes: The author was partially supported by NSF grants DMS 9996078 and 0096946.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.