The computation of Siegel modular forms of weight (2,2) on ( GSp_4 ), particularly those relevant to abelian surfaces over ( \mathbb{Q} ), is indeed challenging due to the non-cohomological nature of this weight, which limits the applicability of cohomological methods like modular symbols. Here's a synthesis of the current state of research and computation:

Key Observations:

1. **Weight (...

Short answer: yes—there is now a substantial amount of explicit computation in the non‑cohomological, weight (2,2) case, but not via the Arthur–Selberg trace formula. The bulk of the data relevant to abelian surfaces over Q comes from computations for the paramodular group K(N), rather than Γ0(N), and has been implemented and tabulated extensively.

What one computes and why

• For principally pola...

Yes – weight (2, 2) (often simply called “weight 2’’) Siegel modular forms of genus 2 have been, and are still being, computed, but not by the same cohomological / modular–symbol method that works so well for weight 2 elliptic modular forms.
The successful calculations use a mixture of

• explicit Fourier–Jacobi expansions,
• Jacobi-form and Gritsenko-lift technology,
• restriction to Humber...

Below is (i) a short description of what is known about scalar weight-2 Siegel cusp forms, (ii) the algorithms that have actually been implemented, and (iii) the data that have already been tabulated. All references are to the classical (genus 2) group
Γ₂ = Sp₄(ℤ)   or to its two most useful congruence subgroups

  Γ₀(N) = { M ∈ Sp₄(ℤ) : c ≡ 0 (mod N) }  

  K(N)   = the “param...

Nobody has ever written down a “black‐box’’ Arthur–Selberg–trace‐formula routine that spits out Fourier–expansions of the (2,2)–forms the way one does for weight 2 on SL₂ by modular symbols. In fact the consensus in the community is

1. For the full Siegel modular group Sp₄(ℤ) the only nonzero cuspidal scalar forms occur in weight ≥10 (Igusa’s χ₁₀, χ₁₂, …) so there is really nothing to compute...

This is an excellent question that goes to the heart of the computational challenges in the Langlands program beyond the classical case. Your intuition about the difficulties is spot on.

Let's break down the situation.

1. Why Weight (2,2) is Hard

You are absolutely correct that the fundamental obstacle is that weight for is not cohomological.

• **For elliptic m...

<think> Okay, the user is asking about computing low-weight Siegel modular forms, specifically weight (2,2), which is challenging because it's not cohomological. They mention that for elliptic curves, modular forms of weight 2 are computed using modular symbols, but that approach fails here. They speculate whether the trace formula could be used instead, but note from FC's comment that this might ...

Computation of Siegel Modular Forms of Weight (2,2)

Your question touches on an important gap in computational number theory. While we have extensive tables for elliptic curves (corresponding to weight 2 modular forms), the situation for abelian surfaces is much less developed.

The computational challenge you've identified is significant: weight (2,2) Siegel modular forms on GSp₄ are non-cohom...

This is an excellent question about a fundamental computational challenge in the theory of Siegel modular forms. Let me address the key issues and what's known about this problem.

The Core Challenge

You're absolutely right that weight (2,2) Siegel modular forms are particularly difficult to compute because this weight is not cohomological. For Siegel modular forms of degree 2, a weight (k,j) ...