Reciprocal divisibility for the stable-base-change period relation

Prove the reciprocal divisibility to the established divisibility between the automorphic periods of a self-dual cuspidal representation of U_E and the middle-degree periods of its stable base change to GL_3(E), equivalently proving the reverse divisibility in the corresponding twisted-adjoint period relation.

Background

The main theorem establishes only one direction of the expected divisibility. The reciprocal direction would yield the full integral period relation up to units and is known in some lower-rank settings, such as quadratic base change for GL_2.

The authors explain that existing proofs rely on non-vanishing modulo p results for suitable twisted L-values or Fourier coefficients. They explicitly note that an analogous non-vanishing result is not known for quasi-split unitary groups.

References

However, to the knowledge of the author, no such result is known for quasi-split unitary groups.

— Stable base change from unitary groups in three variables and integral relation of automorphic periods  (2609.26500 - Ricoul, 22 Sep 2026) in Section 1, subsection “Comments”

In this situation, this is not clear how to define intermediate degree periods as the dimension of the intermediate cohomology groups can be strictly bigger than $2$, whereas the number of involutions available for cutting out $1$-dimensional subspaces remains the same.

— Stable base change from unitary groups in three variables and integral relation of automorphic periods  (2609.26500 - Ricoul, 22 Sep 2026) in Section 1, subsection “Comments”

However, to the knowledge of the author, no such result is known beyond \mathrm{GL}_2.

— 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 “Comments”