Cubic character sums of cubic polynomials
Author:
David J. Wright
Journal:
Proc. Amer. Math. Soc. 100 (1987), 409413
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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 B. Datskovsky and D. J. Wright, The adelic zeta function associated with the space of binary cubic forms, II: Local theory, J. Reine Angew. Math. 367 (1986), 2775. MR 839123 (87m:11034)
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 H. Davenport and H. Hasse, Die Nullstellen der Kongruenz Zetafunktion in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1935), 151182.
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 D. Friedman, Cubic character sums and congruences, Ph.D. Thesis, Univ. of California, Berkeley, 1967.
 [4]
 K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer, 1982. MR 661047 (83g:12001)
 [5]
 J. Lagarias, Evaluation of certain character sums, Master's Thesis, Massachusetts Institute of Technology, 1972.
 [6]
 T. Shintani, On Dirichlet series whose coefficients are classnumbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132188. MR 0289428 (44:6619)
 [7]
 D. J. Wright, The adelic zeta function associated with the space of binary cubic forms, I: Global theory, Math. Ann. 270 (1985), 503534. MR 776169 (86k:11023)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708911363
PII:
S 00029939(1987)08911363
Article copyright:
© Copyright 1987
American Mathematical Society
