Comparing Two Formulas for the Gross-Stark Units
Matthew H. L. Honnor
TL;DR
This work proves, for cubic totally real fields F, that Dasgupta’s explicit p-adic Gross-Stark formula and the cohomological Dasgupta–Spieß formula for the diagonal entries of the Gross regulator matrix are compatible. The authors develop a framework based on Shintani zeta functions, Eisenstein and 1-cocycles, and Colmez domains to control translations of Shintani domains, reducing the problem to explicit calculations within a finite-index rank-2 subgroup V of totally positive units. They execute detailed, explicit calculations (including a counterexample in the appendix to motivate their approach) to show that the two diagonals agree, complementing recent p-adic results of Dasgupta–Kakde. The cubic case thus advances the broader program toward a full diagonal-entailed match and reinforces the interplay between p-adic L-values, regulator matrices, and Shintani-analytic constructions, with implications for principal minors and characteristic polynomials of Gross regulator matrices.
Abstract
Let $F$ be a totally real number field. Dasgupta conjectured an explicit $p$-adic analytic formula for the Gross-Stark units of $F$. In a later paper, Dasgupta-Spiess conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of $F$. Dasgupta-Spiess conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when $F$ is a cubic field.