Geometry of Fermat adeles

Buium, Alexandru

doi:10.1090/S0002-9947-04-03715-8

Geometry of Fermat adeles
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by Alexandru Buium

Trans. Amer. Math. Soc. 357 (2005), 901-964

DOI: https://doi.org/10.1090/S0002-9947-04-03715-8

Published electronically: October 19, 2004

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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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