Arithmeticity at the exceptional full-level value s=0

Prove the arithmeticity of the standard L-function of a full-level Siegel cusp form at s=0 when the degree n satisfies n ≡ 1 mod 4 and n ≠ 1.

Background

The paper proves arithmeticity for nonpositive critical points except when m=0, F=Q, S consists only of the archimedean places, and the twisting character is trivial. This exceptional situation corresponds to the standard L-function of a full-level Siegel cusp form. Earlier work established arithmeticity at s=1, and the functional equation gives arithmeticity at s=0 when n ≡ 3 mod 4. The remaining congruence class n ≡ 1 mod 4, excluding n=1, is explicitly left unresolved.

References

The remaining case is $n \equiv 1 \bmod 4$, whose arithmeticity remains open when $n \neq 1$.

— Arithmeticity of critical $L$-values for Hilbert-Siegel modular forms in the left half plane and $s = 1$  (2610.23037 - Horinaga et al., 16 Oct 2026) in Remark \ref{rem_s=0_except}