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

DOI:
https://doi.org/10.1090/S0002-9939-1987-0891136-3

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.

**[1]**Boris Datskovsky and David J. Wright,*The adelic zeta function associated to the space of binary cubic forms. II. Local theory*, J. Reine Angew. Math.**367**(1986), 27–75. MR**839123**, https://doi.org/10.1515/crll.1986.367.27**[2]**H. Davenport and H. Hasse,*Die Nullstellen der Kongruenz Zetafunktion in gewissen zyklischen Fällen*, J. Reine Angew. Math.**172**(1935), 151-182.**[3]**D. Friedman,*Cubic character sums and congruences*, Ph.D. Thesis, Univ. of California, Berkeley, 1967.**[4]**Kenneth F. Ireland and Michael I. Rosen,*A classical introduction to modern number theory*, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR**661047****[5]**J. Lagarias,*Evaluation of certain character sums*, Master's Thesis, Massachusetts Institute of Technology, 1972.**[6]**Takuro Shintani,*On Dirichlet series whose coefficients are class numbers of integral binary cubic forms*, J. Math. Soc. Japan**24**(1972), 132–188. MR**0289428**, https://doi.org/10.2969/jmsj/02410132**[7]**David J. Wright,*The adelic zeta function associated to the space of binary cubic forms. I. Global theory*, Math. Ann.**270**(1985), no. 4, 503–534. MR**776169**, https://doi.org/10.1007/BF01455301

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0891136-3

Article copyright:
© Copyright 1987
American Mathematical Society