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A generalization of a theorem of Hecke for SL_2(F_p) to fundamental discriminants

Panda, Corina B. (2020) A generalization of a theorem of Hecke for SL_2(F_p) to fundamental discriminants. Journal of Number Theory, 207 . pp. 83-109. ISSN 0022-314X. https://resolver.caltech.edu/CaltechAUTHORS:20190523-160733231

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Abstract

Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π+, π− be the pair of cuspidal representations of SL_2(F_p). It is well known by Hecke that the difference m_π+ −m_π− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(√−p). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q(√−D) and prove that an alternating sum of multiplicities of certain irreducibles of SL_2(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 ≡ 0 mod 8. The proof uses the holomorphic Lefschetz number.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jnt.2019.04.009DOIArticle
Alternate Title:A generalization of a theorem of Hecke for SL2(Fp) to fundamental discriminants
Additional Information:© 2019 Elsevier Inc. Received 5 August 2016, Revised 15 October 2018, Accepted 28 April 2019, Available online 23 May 2019.
Subject Keywords:Hecke; Class number; Ffundamental discriminant; Imaginary quadratic field; Lefschetz number; Fixed points; Modular curve
Record Number:CaltechAUTHORS:20190523-160733231
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190523-160733231
Official Citation:Corina B. Panda, A generalization of a theorem of Hecke for SL2(Fp) to fundamental discriminants, Journal of Number Theory, Volume 207, 2020, Pages 83-109, ISSN 0022-314X, https://doi.org/10.1016/j.jnt.2019.04.009. (http://www.sciencedirect.com/science/article/pii/S0022314X19301490)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:95772
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 May 2019 23:17
Last Modified:15 Oct 2019 16:36

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