Comparing Two Formulas for the Gross-Stark Units

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.

💡 Research Summary

The paper addresses a conjectural compatibility between two distinct formulas for Gross‑Stark units in the setting of a totally real number field F of degree three. The first formula, proposed by Dasgupta, gives an explicit p‑adic analytic expression for the Gross‑Stark unit in terms of values at s = 0 of Shintani zeta functions attached to certain Shintani domains. The second formula, due to Dasgupta and Spiess, arises from a cohomological construction: the diagonal entries of the Gross regulator matrix are expressed as the ratio of the p‑adic logarithm to the p‑adic order of the χ‑1 component of the Gross‑Stark unit, where χ is a totally odd character of Gal(H/F) and H is the CM field cut out by χ.

The main theorem (Theorem 6.3) proves that, when F is a cubic field, these two expressions coincide. The proof proceeds by a careful analysis of Shintani sets, which are subsets of the positive orthant ℝ³⁺ constructed from totally positive elements of F. The unit group E⁺(𝔣) (totally positive units congruent to 1 modulo the conductor 𝔣 of H/F) acts on ℝ³⁺, and a Shintani domain D is a fundamental domain for this action. However, directly controlling the translates of D by elements of E⁺(𝔣) is difficult because there is no a priori bound on how far a translate can move D.

To overcome this, the author replaces the full unit group by a finite‑index free subgroup V ⊂ E⁺(𝔣) of rank two. A fundamental domain for V is called a Colmez domain D_V. By invoking results of Colmez, the author selects generators ε₁, ε₂ of V that satisfy a sign condition: for every permutation τ of {1,2}, the determinant sign δ(