A generalization of a theorem of Hecke for SL2(Fp) to fundamental discriminants

JOURNAL OF NUMBER THEORY

卷:207

Article

Panda, Corina B.¹ 

[1] CALTECH, Dept Math, MC 253-37, Pasadena, CA 91125 USA

关键词: Hecke;    Class number;    Fundamental discriminant;    Imaginary quadratic field;    Lefschetz number;    Fixed points;    Modular curve;   

DOI  :  10.1016/j.jnt.2019.04.009

来源: Elsevier

【 摘 要 】

Let p > 3 be an odd prime, p equivalent to 3 mod 4 and let pi(+), pi(-) be the pair of cuspidal representations of SL2(F-p). It is well known by Hecke that the difference m(pi+) - m(pi-) in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Gamma(p), equals the class number h(-p) of the imaginary quadratic field Q(root-p). We extend this result to all fundamental discriminants -D of imaginary quadratic fields Q(root-D) and prove that an alternating sum of multiplicities of certain irreducibles of SL2(Z/DZ) is an explicit multiple, up to a sign and a power of 2, of either the class number h(-D) or of the sums h(-D) h(-D/2), h(-D)+2h(-D/2); the last two possibilities occur in some of the cases when D equivalent to 0 mod 8. The proof uses the holomorphic Lefschetz number. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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