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

Gundlach, Karl-Bernhard

Glasgow mathematical journal, Tome 26 (1985), pp. 57-80

Résumé

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} byNowadays 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 formJ(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).

Détail

DOI : 10.1017/S0017089500006078

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