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Multiplier systems for Hilbert's and Siegel's modular groups

Published online by Cambridge University Press:  18 May 2009

Karl-Bernhard Gundlach
Karl-Bernhard Gundlach
Affiliation:
Fachbereich Mathematik Der Universität Marburg Lahnberge, 3550 Marburg/Lahn, Germany
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The classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK = SL(2, o)(o the principal order of a totally real number field K, [K:ℚ]=n), operating, originally, on a product of n upper half-planes or, for n=2, on the product 1×− of an upper and a lower half-plane by

(where v(i), for v∈K, denotes the jth conjugate of v), and Гn = Sp(n, ℤ), operating on n={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} by

Nowadays ГK is called Hilbert's modular group of K and Гn Siegel's modular group of degree (or genus) n. For n=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the form

J(L, t) for fixed L (or J(M, Z) for fixed M) denoting a holomorphic function without zeros on ) (or on n). A function J;, defined on ℤK×or ℤn×n to be able to appear in (1.3) with f≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the case n=1, mainly AFs of the following kind have been used:

with a complex number r, the weight of J, and complex numbers v(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. If r∉ℤ, the values of the function v on ГK (or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for each L∈ГK or M∈Гn) the functional equations for J, by (1.4), (1.5), correspond to functional equations for v. A function v satisfying those equations is called a multiplier system (MS) of weight r for ГK (or Гn).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 57 - 80
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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