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The Reciprocity of Dedekind Sums and the Factor Set for the Universal Covering Group of SL(2, R)

Published online by Cambridge University Press:  22 January 2016

Tetsuya Asai
Tetsuya Asai*
Affiliation:
Mathematical Institute, Nagoya University
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By explicit studying on theta-multipliers (i.e. the multipliers which appear in theta-transformation formulas under general modular substitutions), we can naturally get the reciprocity law of Gauss sums or quadratic residue symbols. This remarkable fact, by Cauchy, Kronecker, Hecke and others, is very classical, but its theoretical meaning has not been sufficiently clear yet.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 37 , March 1970 , pp. 67 - 80
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1970

References

[1] Dedekind, R., Erläuterung zu den Fragmenten XXVIII, in Riemann, B., Ges. Math. Werke, Leipzig, 1892, SS. 466–478; Dedekind’s Ges. Werke I, 1930, SS. 159–172.Google Scholar
[2] Dieter, U., Beziehungen zwischen Dedekindschen Summen, Abh. Math. Sem. Hamburg, 21 (1957), SS. 109–125.CrossRefGoogle Scholar
[3] Kubota, T., Reciprocity laws and automorphic functions, Sûgaku, 18 (1966), pp. 10–24 (in Japanese)*).Google Scholar
[4] Kubota, T., Topological covering of SL(2) over a local field, J. Math. Soc. Japan, 19 (1967), pp. 114–121.CrossRefGoogle Scholar
[5] Petersson, H., Zur analytischen Theorie der Grenzkreisgruppen I, Math. Ann., 115 (1938), SS. 23–67.CrossRefGoogle Scholar
[6] Rademacher, H., Zur Theorie der Modulfunktionen, J.f.d. reine u. angew. Math., 167 (1931), SS, 312–336.Google Scholar
[7] Rademacher, H., Über die Transformation der Logarithmen der Thetafunktionen, Math. Ann., 168 (1967), SS. 142–148.CrossRefGoogle Scholar
[8] Rademacher, H. and Whiteman, , Theorems on Dedekind sums, Amer. J. Math., 63 (1941), pp. 377–407.CrossRefGoogle Scholar