Periods of Hecke Characters
The starting point of this Lecture Notes volume is Deligne's theorem about absolute Hodge cycles on abelian varieties. Its applications to the theory of motives with complex multiplication are systematically reviewed. In particular, algebraic relations be
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1301 Norbert Schappacher
Periods of Heeke Characters
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Norbert Schappacher Max-Planck-Institut fur Mathematik Gottfried-Claren-StraBe 26 5300 Bonn 3, Federal Republic of Germany
Mathematics Subject Classification (1980): Primary: 10D25, 14A20, 14K22, 12C20 Secondary: 14C99, 18F99, 14G 10, 12C25, 33A 15, 14K20 ISBN 3-540-18915-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387 -18915-7 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
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INTRODUCTION
In two papers
n" 12 and 14 of [He], published in 1918 and 1920
introduced what he called
E. Heeke
of algebraic number fields, with
a view to extending the theory of L-functions and their applications in analytic number theory. In the early 1950's the arithmetic and geometric significance of those of Hecke's characters that take algebraic values began to appear in two different, if overlapping, lines of thought. (Both of these had been anticipated in special cases by Eisenstein exactly one hundred years earlier; but none of the mathematicians working on them in the fifties seems to have been aware of their precursor at the time.) First Weil, testing a conjecture of Hasse, investigated algebraic curves over Q with the property that the number of F p rational points on their reductions modulo p can be computed in terms of exponential sums. This led him to a study of "Jacobi sums as 'Gro,Bencharaktere' ". Secondly Deuring, developing one aspect of Weil's examples, proved that the (HasseWeil) L-function of an elliptic curve with complex multiplication is a (product of) Heeke L-function(s). This was then quickly generalized to higher dimensional C'M abelian varieties by Shimura and Taniyama, with Weil providing clarification, for instance, on the Heeke characters employed in the theory. Both approaches cover only very limited classes of algebraic Heeke characters. Jacobi sum characters were confined to cyclotomic (today: abelian) fields, and in general, not every algebraic Heeke character of such a field is given by Jacobi sums. - The product of several Heeke characters each one of which is attached to a eM abelian variet
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