Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On zeta elements and functional equations for Tate motives over totally real fields

Mahiro Atsuta

TL;DR

The paper advances the Iwasawa-theoretic understanding of Tate motives over totally real fields by constructing zeta elements that interpolate $L$-values at positive integers under an unramified-at-$p$ condition and by establishing compatibility with Tate twists, thereby providing partial verification of the generalized Iwasawa main conjecture. It develops a global-local determinant framework, defines generalized Stark elements $oldsymbol{ exteta}_{L,S}^j$ in exterior biduals, and proves key properties linking Fitting ideals and regulator images to $L$-values, as well as a congruence across twists that echoes Burns–Kurihara–Sano predictions. The work also connects to the minus ETNC proved by Dasgupta–Kakde–Silliman and recovers cyclotomic-unit phenomena in the base field $b Q$, yielding an Euler-system perspective for the rank-$r_k$ case. Overall, it lays out a structured pathway from ETNC and local duality results to global Euler systems in the setting of Tate motives over totally real fields. The results thus provide a concrete, twist-compatible framework for relating special $L$-values, determinant modules, and higher-rank Euler systems in Iwasawa theory.

Abstract

In this paper, we study Iwasawa theory for Tate motives over totally real fields. More precisely, we construct a zeta element that interpolates the values of $L$-functions at positive integers over totally real fields under a certain unramified condition at $p$. As an application of this, we construct a canonical element in the exterior power bidual of the Galois cohomology group that is also related to the values of $L$-functions at positive integers.

On zeta elements and functional equations for Tate motives over totally real fields

TL;DR

The paper advances the Iwasawa-theoretic understanding of Tate motives over totally real fields by constructing zeta elements that interpolate -values at positive integers under an unramified-at- condition and by establishing compatibility with Tate twists, thereby providing partial verification of the generalized Iwasawa main conjecture. It develops a global-local determinant framework, defines generalized Stark elements in exterior biduals, and proves key properties linking Fitting ideals and regulator images to -values, as well as a congruence across twists that echoes Burns–Kurihara–Sano predictions. The work also connects to the minus ETNC proved by Dasgupta–Kakde–Silliman and recovers cyclotomic-unit phenomena in the base field , yielding an Euler-system perspective for the rank- case. Overall, it lays out a structured pathway from ETNC and local duality results to global Euler systems in the setting of Tate motives over totally real fields. The results thus provide a concrete, twist-compatible framework for relating special -values, determinant modules, and higher-rank Euler systems in Iwasawa theory.

Abstract

In this paper, we study Iwasawa theory for Tate motives over totally real fields. More precisely, we construct a zeta element that interpolates the values of -functions at positive integers over totally real fields under a certain unramified condition at . As an application of this, we construct a canonical element in the exterior power bidual of the Galois cohomology group that is also related to the values of -functions at positive integers.
Paper Structure (28 sections, 19 theorems, 161 equations)

This paper contains 28 sections, 19 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments
  4. Preliminary
  5. Determinant module
  6. Algebraic notation
  7. Galois cohomology complex
  8. Notation
  9. Local Galois cohomology complex
  10. Global cohomology complex
  11. Compatibility of the isomorphisms $\vartheta_{K/k, S}^j$ and $\Phi_{K/k, S}^{\mathop{\mathrm{loc}}\nolimits, j}$
  12. Iwasawa theory over a totally real field
  13. The $L$-functions
  14. The minus component of the equivariant Tamagawa number conjecture
  15. Zeta element for $\mathbb{Z}_p (j)$ when $j \leq 0$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Assume that $k / \mathbb{Q}$ is unramified at $p$ and $K/k$ is also unramified at all $p$-adic primes. Then, for any $j \in \mathbb{Z}_{\geq 1}$, there is a (unique) $\varepsilon_j \Lambda$-basis such that the following statements hold.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 35 more