A Caltech Library Service

Generalizations of a Theorem of Hecke


Panda, Corina Bianca (2020) Generalizations of a Theorem of Hecke. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/bxgs-4825.


Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π⁺, π⁻ be the pair of cuspidal representations of SL₂(𝔽p). It is well known by Hecke that the difference mπ⁺ - mπ⁻ 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)).

This thesis consists of two main parts. In the first part, we extend Hecke's result to all fundamental discriminants of imaginary quadratic fields, including the even case. The proof is geometric in nature and uses the holomorphic Lefschetz number.

In the second part, we consider generalizations to groups with higher ℚ-rank. In particular, we focus on the rank 2 special unitary group SU(2, 2). On the representation theory side, we prove the regular unipotent classes have positive contribution to an alternating sum of multiplicities of certain irreducible cuspidal representations of SU(2, 2) over the finite field of p elements. We also show that the semisimple classes have zero contribution, which is again a direct generalization of the SL₂ case. To obtain these two results, we make use of the Deligne-Lusztig theory and the connection of the traces to the Gelfand-Graev representations.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Hecke; class number; Lefschetz number; Deligne-Lusztig representations; Gelfand-Graev representations; Gauss sums; toroidal compactification
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Apostol Award for Excellence in Teaching in Mathematics, 2016-2016.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Mantovan, Elena (chair)
  • Ramakrishnan, Dinakar
  • Graber, Thomas B.
  • Yom Din, Alexander
Defense Date:1 August 2019
Record Number:CaltechTHESIS:06072020-124705473
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 2.
Panda, Corina Bianca0000-0002-6637-211X
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13784
Deposited By: Corina B. Panda
Deposited On:09 Jun 2020 18:01
Last Modified:16 Jun 2020 21:16

Thesis Files

PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page