Class numbers and invariant characters of \mathfraksl₂(\mathbbF_p)

2025/08/24 by Chen, Zhe, Feng, Yongqi
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2508.17214

Abstract

Let p be a prime and let S₂(Γ(p)) be the space of weight 2 cusp forms for the principal congruence subgroup Γ(p). Then SL₂(\mathbbF_p) acts on S₂(Γ(p)) in a natural way. Around 1928, Hecke proved that if p>3 and p≡ 3\mod 4, then the class number of ℚ(√(-p)) is equal to the difference between the multiplicities of two particular irreducible representations of SL₂(\mathbbF_p) in S₂(Γ(p)). In this paper we prove a Lie algebra analogue of this result. As an application we extend Hecke's result to SL₂(ℤ/p^r) (acting on S₂(Γ(p^r))) for any r≥ 2.

Citations

• The algebraisation of higher level Deligne--Lusztig representations II: odd levels
• The algebraisation of higher Deligne--Lusztig representations
• A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications

Related

• A note on cusp forms and representations of SL₂(\mathbbF_p)
• On the dimensions of certain spaces of vector-valued cusp forms
• The invariants of the second symmetric power representation of SL₂(F_q)
• A look at Representations of SL(2,F_q) through the Lens of Size
• Period polynomials and explicit formulas for Hecke operators on Γ₀(2)