Values of the log-rigid class equal p-adic logarithms of Gross–Stark units

Establish that for any totally real field F of degree n in which the rational prime p is inert, and for any integral ideal a of OF with norm coprime to pc, if T ∈ Fn is a column vector whose entries form an oriented Z-basis of a^{-1} and hence determine a point T ∈ Xp = P^{n-1}(Cp) \ ⋃_{H} H (Drinfeld’s p-adic symmetric domain), then the value of the log-rigid analytic cohomology class JE,c ∈ H^{n−1}(SLn(Z), Ac) at T satisfies JE,c[T] = log_p(u^{σ_a}), where u ∈ OH[1/p]^ is the Gross–Stark unit in the narrow Hilbert class field H of F attached to p and c, and σ_a ∈ Gal(H/F) is the Frobenius element corresponding to the ideal a.

Background

The paper constructs a log-rigid analytic cohomology class JE,c for SLn(Z) by starting from the Eisenstein class of a torus bundle and applying a p-adic Poisson kernel to obtain a class valued in log-rigid analytic functions on Drinfeld’s symmetric domain Xp. For a totally real field F of degree n with p inert, a special point T ∈ Xp is defined from an oriented Z-basis of an ideal a^{-1}, and the class JE,c is evaluated at T to produce a value in Fp.

They prove that the local trace Tr_{Fp/Qp}(JE,c[T]) matches the derivative at s=0 of a p-adic partial zeta function and thus equals Tr_{Fp/Qp}(log_p(u^{σ_a})) for a Gross–Stark unit u (Theorem 1.5), and they establish the full equality JE,c[T] = log_p(u^{σ_a}) in certain Galois situations (Theorem 1.7). Motivated by these results and by the n=2 case of Darmon–Pozzi–Vonk, they conjecture that the equality holds generally without taking traces, providing a modular-like construction of Gross–Stark units via the values of JE,c.

Proving this conjecture would yield a direct, automorphic construction of Gross–Stark units across totally real fields, bridging Eisenstein cohomology, p-adic L-functions, and explicit class field theory. It generalizes known results in degree n=2 and would complete the link between the topological Eisenstein class framework and arithmetic units.