On the Equality of Three Formulas for Brumer--Stark Units

We prove the equality of three conjectural formulas for the Brumer–Stark units. The first formula has essentially been proven, so the present paper also verifies the validity of the other two formulas.

💡 Research Summary

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The paper establishes that three conjectural formulas for Brumer–Stark units—denoted u₁, u₂, and u₃—are in fact equal in the appropriate tensor product space. The setting is a totally real number field F, a finite abelian extension H/F, and a prime p that splits completely in H. The Brumer–Stark conjecture predicts the existence of a p‑unit u_T in H satisfying certain norm and valuation conditions expressed via partial zeta‑values ζ_{R,T}(H/F,σ,0). Collecting the Galois conjugates of u_T yields the Brumer–Stark element u_p ∈ H^× ⊗ ℤ