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Evaluating symplectic Gauss sums and Jacobi Symbols

Published online by Cambridge University Press:  22 January 2016

Robert Styer*
Department of Mathematics, Villanova University, Villanova, PA 19085, USA
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Stark [9] has explicitly evaluated some symplectic Gauss sums when the “denominator” matrix has odd prime level. This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]).

Research Article
Copyright © Editorial Board of Nagoya Mathematical Journal 1984


[ 1 ] Andrianov, A. N. and Maloletkin, G. N., Behavior of theta series of degree n under modular substitutions, Math USSR-Izv., 9 (1975), No. 2, 227241.CrossRefGoogle Scholar
[ 2 ] Eichler, Martin, Introduction to the Theory of Algebraic Numbers and Functions, Academic Press, New York, 1966, 4652.Google Scholar
[ 3 ] Friedberg, Solomon, Theta series, the Weil representation, and the corresponding eighth root of unity, preprint.Google Scholar
[ 4 ] Hasse, Helmut, Number Theory, Springer-Verlag, Berlin, 1980, p. 86.CrossRefGoogle Scholar
[ 5 ] Hecke, Erich, Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, New York, 1948, 218249.Google Scholar
[ 6 ] Lang, Serge, Algebraic Number Theory, Addison-Wesley, Reading, Mass. 1970, p. 64.Google Scholar
[ 7 ] Maass, Hans, SiegePs Modular Forms and Dirichlet Series, Lecture Notes in Mathematics, 216, Springer-Verlag, Berlin, 1971, 155164.Google Scholar
[ 8 ] Newman, Morris, Integral Matrices, Pure and Applied Mathematics, 45, Academic Press, New York, 1972.Google Scholar
[ 9 ] Stark, H. M., On the transformation formula for the symplectic theta function and applications, J. Fac. Sci., Univ. of Tokyo, 29 (1982), No. 1, 112.Google Scholar
[10] Stark, H. M., Some examples of modular forms over number fields. I. to appear.Google Scholar
[11] Styer, Robert, Prime determinant matrices and the symplectic theta function, Amer. J. Math., 106 #3 (1983), 645665.CrossRefGoogle Scholar
[12] Weber, H., Ueber die mehrfachen Gaussischen Summen, J. Reine Angew. Math., 74 (1872), 1456.CrossRefGoogle Scholar