On the Root of Unity Ambiguity in a Formula for the Brumer--Stark Units

TL;DR

The paper resolves the root-of-unity ambiguity in the Brumer–Stark unit formula by establishing an exact equality between the cohomological u2-formula and the Brumer–Stark unit u_p, removing prior S-based hypotheses. It leverages the integral Gross–Stark conjecture (p-part proven by Dasgupta–Kakde and lattice-level results via eTNC^−) to connect Stickelberger data with Brumer–Stark units, first reducing the ambiguity to 2-power roots and then eliminating it completely through augmentation and norm arguments. By proving u1 = u2 = u3 and removing the last obstructions, the work solidifies explicit Brumer–Stark unit computations and strengthens ties to Iwasawa-theoretic methods. Overall, it confirms the exact Brumer–Stark equality over Z for all p-split abelian extensions, without room for nontrivial root-of-unity ambiguity.

Abstract

We prove a conjectural formula for the Brumer--Stark units. Dasgupta--Kakde have shown the formula is correct up to a bounded root of unity. In this paper we resolve the ambiguity in their result. We also remove an assumption from Dasgupta--Kakde's result on the formula.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Conjecture 2.3: Conjecture $7.4$, MR931448
• Proposition 2.4: Proposition 6.3, MR3861805
• Conjecture 2.5: Conjecture 6.1, MR3861805
• Theorem 2.6: Theorem 1.6, intgrossstark
• Theorem 2.7
• Theorem 3.1
• Lemma 3.2
• proof