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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Imprimitive Gaussian sums and theta functions over a number field
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by Jacob Nemchenok
Trans. Amer. Math. Soc. 338 (1993), 465-478
DOI: https://doi.org/10.1090/S0002-9947-1993-1041052-9
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Abstract:

We obtain a reduction formula for an imprimitive Gaussian sum with a numerical character in an algebraic number field, i.e. a formula that expresses that sum as a product of several elementary factors times a primitive, proper, normed Gaussian sum (formulae (16) and (19)). We also introduce Gaussian sums with Hecke characters and derive a similar reduction formula for them. The derivation is based on an inversion formula for a multivariable theta function associated with the number field, twisted with the numerical character.
References
    H. Hasse, Vorlezungen über Zahlentheorie, zweite auf., Springer-Verlag, Berlin, 1964. —, Allgemeine Theorie der Gaussche Summen in algebraischen Zahlkörpern, Mathematische Abhandlungen, Vol. 3, De Gruyter, Berlin and New York, 1975, pp. 15-39.
  • E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), no. 1-2, 11–51 (German). MR 1544392, DOI 10.1007/BF01202991
    • —, Vorlezungen über die Theorie der algebraischen Zahlen, Akademische Verlag, 1923; English transl., Lectures on the theory of algebraic numbers, Springer-Verlag, 1981.
  • Henri Joris, On the evaluation of Gaussian sums for non-primitive Dirichlet characters, Enseign. Math. (2) 23 (1977), no. 1-2, 13–18. MR 441888
  • Erich Lamprecht, Allgemeine Theorie der Gaussschen Summen in endlichen kommutativen Ringen, Math. Nachr. 9 (1953), 149–196 (German). MR 54578, DOI 10.1002/mana.19530090303
  • David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
  • Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
  • R. Odoni, On Gauss sums $(\textrm {mod}\ p^{n})$, $n\geq 2$, Bull. London Math. Soc. 5 (1973), 325–327. MR 327678, DOI 10.1112/blms/5.3.325
  • H. M. Stark, Modular forms and related objects, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 421–455. MR 894333
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Bibliographic Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 338 (1993), 465-478
  • MSC: Primary 11L05; Secondary 11F11, 11F12
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1041052-9
  • MathSciNet review: 1041052