Remove the inert-place local assumption in Bellaïche–Chenevier’s unitary-group results

Establish that the strong local assumption imposed at inert places in Bellaïche–Chenevier’s work can be removed when the relevant place v does not divide 2, provided that the eigenvarieties used in their main results interpolate the eigenvalues of the Hecke operators constructed in this paper.

Background

The paper develops Hecke operators intended to control the ramification and monodromy of Galois representations in p-adic families for unitary and other classical groups. The authors explain that Bellaïche–Chenevier impose a strong local hypothesis at inert places in their treatment of unitary groups. The newly constructed operators could potentially provide the missing local control needed to dispense with that hypothesis, but the paper does not establish that the relevant eigenvarieties interpolate these operators or that the resulting assumption can indeed be removed.

References

Presumably one can remove this condition there as well when $v\nmid 2$, as long as the eigenvarieties whose construction is relevant for their main results can be made to interpolate the eigenvalues of the operators $\phi_v$ constructed here.

— Degeneration of monodromy in the theory of Eisenstein congruences for classical groups  (2609.04703 - Mundy, 4 Sep 2026) in Introduction, subsection “Past work and future applications”