Triviality of the ramified local factor

Prove that the local-factor constant $u_{E/F}$ appearing in the adjoint L-value formula for the quasi-split unitary group U_{E/F} is equal to $1$ for every totally real field F and quadratic extension E/F satisfying the hypotheses of the formula.

Background

The adjoint L-value formula contains a product of normalized local Whittaker factors at places where E/F is ramified. This product is denoted by uE/Fu_{E/F}. The authors prove that it belongs to the rationality field of the finite part of the automorphic representation and establish uE/Q=1u_{E/\mathbf Q}=1 when F=Q.

For general totally real F, the formula states that the constant is conjectured to be 1, but the paper does not compute the remaining ramified local factors in full generality.

References

where $u_{E/F} \in Q(\pi_f)\times$ is a constant conjectured to be $1$. For $F=Q$, one has that $u_{E/Q} =1$.

— Stable base change from unitary groups in three variables and integral relation of automorphic periods  (2609.26500 - Ricoul, 22 Sep 2026) in Section 4, subsection “Triviality and algebraicity of $u_{E/F}$”; Theorem 1.4

The constant u is expected to be 1 (and hence S_u is expected to be empty) but is hard to compute because it involves local linear forms which are defined indirectly and have no explicit formula.

— A divisibility of automorphic periods for the real quadratic base change of $\mathrm{GL}_3$  (2609.26477 - Ricoul, 22 Sep 2026) in Section 1, subsection “The Bloch–Kato conjecture for the twisted adjoint motive of \(\pi\)”; discussion following Theorem B (thmB) and Section 4, subsection “Algebraicity of uncomputed local factors”