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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cubic character sums of cubic polynomials


Author: David J. Wright
Journal: Proc. Amer. Math. Soc. 100 (1987), 409-413
MSC: Primary 11T21; Secondary 11E45, 11E76, 11L10
MathSciNet review: 891136
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Abstract: A complete evaluation is given of the sum over an arbitrary finite field of the values of a nontrivial cubic character of the field applied to an arbitrary polynomial of degree not greater than three in one variable defined over the field. Previous evaluations for the fields of prime order were given in theses of Friedman and Lagarias. The evaluation given below makes simple use of standard facts about equivalence of binary cubic forms. An interesting consequence of this evaluation is given connecting the values of these sums over the space of all polynomials of degree not greater than three over the finite field. The evaluation of these sums is of relevance to the theory of Shintani's Dirichlet series associated to the space of binary cubic forms since they appear in the residues of the cubic twists of these series.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0891136-3
PII: S 0002-9939(1987)0891136-3
Article copyright: © Copyright 1987 American Mathematical Society