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

Samit Dasgupta, Matthew H. L. Honnor, Michael Spieß

TL;DR

The paper proves the equivalence of three conjectural Brumer–Stark unit formulas $u_1,u_2,u_3$, tying a p-adic analytic construction to two cohomological realizations. It leverages Shintani zeta functions, Colmez subgroups, and Eisenstein cocycles to define $u_2$ and $u_3$, then shows $u_2=u_3$ via a functorial cap-product framework and $u_1=u_3$ through a norm-compatibility argument that handles CM and non-CM cases and addresses edge cases with auxiliary primes. The result validates the broader Brumer–Stark program and connects to explicit class field theory and conjectures on the Gross–Regulator matrix. Methodologically, it synthesizes adelic/reciprocity data, Shintani domains, and higher algebraic techniques to bridge analytic and cohomological viewpoints on Stark-type units.

Abstract

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.

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

TL;DR

The paper proves the equivalence of three conjectural Brumer–Stark unit formulas , tying a p-adic analytic construction to two cohomological realizations. It leverages Shintani zeta functions, Colmez subgroups, and Eisenstein cocycles to define and , then shows via a functorial cap-product framework and through a norm-compatibility argument that handles CM and non-CM cases and addresses edge cases with auxiliary primes. The result validates the broader Brumer–Stark program and connects to explicit class field theory and conjectures on the Gross–Regulator matrix. Methodologically, it synthesizes adelic/reciprocity data, Shintani domains, and higher algebraic techniques to bridge analytic and cohomological viewpoints on Stark-type units.

Abstract

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.
Paper Structure (24 sections, 35 theorems, 234 equations)

This paper contains 24 sections, 35 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Preliminaries for the multiplicative integral formula
  3. Notation
  4. Partial zeta functions
  5. Shintani zeta functions
  6. The multiplicative integral formula ($u_1$)
  7. Transferring to a subgroup
  8. Preliminaries for the cohomological formulas
  9. Continuous maps
  10. Measures
  11. Eisenstein cocycles
  12. Colmez subgroups
  13. 1-cocycles attached to homomorphisms
  14. Homology of a group of units
  15. Cohomological formula I ($u_2$)
  16. ...and 9 more sections

Key Result

Theorem 1.3

We have the following equalities between the three conjectural formulas for the Brumer--Stark element $u_\mathfrak{p}$, namely,

Theorems & Definitions (79)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4: Theorem 1.6, intgrossstark
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 69 more