[Paper Review] The trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two

Rainer Weissauer|ArXiv.org|Sep 9, 2009

Advanced Algebra and Geometry9 references16 citations

TL;DR

This paper computes the trace of Hecke operators on the space of classical holomorphic Siegel modular cusp forms of genus two for the full Siegel modular group, using the trace formula and automorphic representation theory. The key result expresses the trace in terms of Frobenius traces on étale cohomology, Eisenstein cohomology, and Galois representations attached to elliptic cusp forms, confirming a conjecture of [FG] for regular weights.

ABSTRACT

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

Motivation & Objective

• To compute the trace of Hecke operators on the space of holomorphic Siegel cusp forms of genus two for the full Siegel modular group.
• To relate this trace to the cohomology of Shimura varieties and Galois representations via the trace formula.
• To confirm Conjecture 4.1 of [FG] on the Euler characteristic of automorphic cohomology for regular weights.
• To decompose the cohomology of the moduli space A₂ into components associated with Eisenstein, CAP, endoscopic, and non-CAP cusp forms.

Proposed method

• Applies the trace formula from [W1]–[W3] to the group GSp(4,𝔸) to analyze the trace of Hecke operators on Siegel cusp forms.
• Uses the decomposition of the cohomology H₆c(A₂, V_λ) into Eisenstein, CAP, endoscopic, and non-CAP (00) components.
• Relies on the Eichler-Shimura isomorphism to relate cohomology of A₁ to elliptic modular forms of weight r₂.
• Uses the identification of cohomological cuspidal automorphic representations with holomorphic discrete series of weight (k₁,k₂) for k₁ ≥ k₂ ≥ 3.
• Applies the theory of weak endoscopic lifts and CAP representations to isolate the contribution of non-CAP forms.
• Employs the trace formula to express the trace of T(p) as a combination of Frobenius traces on cohomology and Galois representations.

Experimental results

Research questions

• RQ1 How can the trace of Hecke operators on genus two Siegel cusp forms be computed using automorphic and Galois-theoretic methods?
• RQ2 What is the precise decomposition of the cohomology of the moduli space A₂ into Eisenstein, CAP, endoscopic, and non-CAP components?
• RQ3 Does the trace formula confirm Conjecture 4.1 of [FG] for the Euler characteristic of automorphic cohomology in regular weights?
• RQ4 What is the role of Galois representations attached to elliptic cusp forms in the trace of Hecke operators on genus two Siegel cusp forms?
• RQ5 How does the cohomology of the Shimura variety M decompose under the action of GSp(4,𝔸_f) for regular weights?

Key findings

• The trace of the Hecke operator T(p) on the space of Siegel cusp forms of weight (k₁,k₂) with k₁ > k₂ > 3 is given by a formula involving the trace of Frobenius on cohomology, Galois representations, and Eisenstein contributions.
• For regular weights k₁ > k₂ > 3, the Eisenstein cohomology H_E^•(M,V_λ) vanishes, simplifying the trace formula.
• The non-CAP cusp cohomology component H_00^•(M,V_λ) contributes a motif of rank 4·dim_ℂ([Γ₂,ρ]_00), confirming the motivic structure predicted by [FG].
• The formula for the trace of T(p) includes a term involving the trace of Frobenius on the tensor product of Galois representations ρ₂ and the character ν_l^{k₂−2}, weighted by the dimension of the space of elliptic cusp forms of weight r₁.
• The Euler characteristic e_c(A₂, V_λ) decomposes into e_Eis, e_E, e_endo, and e_00 components, with e_00 corresponding to the non-CAP cusp forms.
• The result confirms Conjecture 4.1 of [FG] by showing that the trace formula matches the predicted L-function behavior for regular weights.