Multiplier systems for Siegel modular groups

Abstract

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus $g>1$ must be integral or half integral. Actually he proved that for a system $v(M)$ of complex numbers of absolute value 1

$$\begin{array}{ll}\hfill v(M)det{(CZ+D)}^r(r\in ℝ)(0.1)& \end{array}$$

can be an automorphy factor only if $2r$ is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann.159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for $S{L}_n(n\ge 3)$ and $S{p}_{2n}(n\ge 2)$, Publ. Math. Inst. Hautes Études Sci.33 (1967) 59–137] of Bass–Milnor–Serre.