Integral period relation for stable base change

Establish the conjectural integral relation between the middle-degree minus period of the stable base change representation and the product of the bottom- and top-degree automorphic periods of the self-dual representation of the quasi-split unitary group, including the congruence-defect factor.

Background

The paper introduces middle-degree automorphic periods for self-conjugate cuspidal representations of GL_3 over a real quadratic field because the stable base change from the quasi-split unitary group U_E naturally corresponds to degree-5 cohomology. The authors expect these periods, rather than the extremal top- and bottom-degree periods, to enter the integral period relation.

The proposed relation compares the minus middle-degree period of the stable base change Π=SBC(π_U) with the product of the degree-2 and degree-3 periods of π_U, multiplied by a factor measuring the difference between Hecke and cohomological congruence numbers. The paper proves only one divisibility in the predicted relation.

References

This leads us to formulate the following conjectural integral relation between the periods of $\pi_U$ and the newly defined middle-degree periods of $\Pi$ (see for a precise motivation) :

— 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 “A conjectural integral period relation for the stable base change,” Main Conjecture (1.1)

We now assume that the following three conjectures hold :

— Stable base change from unitary groups in three variables and integral relation of automorphic periods  (2609.26500 - Ricoul, 22 Sep 2026) in Section 2, subsection “Relation with top and bottom degree periods”

This leads us to formulate the following conjectural integral relation between the periods of \pi and the newly defined middle-degree periods of \Pi (see for a precise motivation):

— 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 “A conjectural integral period relation for the real quadratic base change of GL_3”; Main Conjecture (main_conj)