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Wall crossing in Iwasawa theory

Shilin Lai

Abstract

This paper sets up a framework to organize anticyclotomic Iwasawa theory in the context of the Gan-Gross-Prasad conjecture for unitary groups. We propose multiple main conjectures depending on archimedean weight interlacing conditions, generalizing phenomena in the anticyclotomic Iwasawa theory of elliptic curves. We also prove an abstract theorem in Galois cohomology which relates the conjectures.

Wall crossing in Iwasawa theory

Abstract

This paper sets up a framework to organize anticyclotomic Iwasawa theory in the context of the Gan-Gross-Prasad conjecture for unitary groups. We propose multiple main conjectures depending on archimedean weight interlacing conditions, generalizing phenomena in the anticyclotomic Iwasawa theory of elliptic curves. We also prove an abstract theorem in Galois cohomology which relates the conjectures.

Paper Structure

This paper contains 38 sections, 10 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related and future works
  4. Diagonal condition
  5. Other conditions
  6. Notations and conventions
  7. p-adic family of Galois representations
  8. Weight space
  9. p-adic family
  10. Commutative algebra
  11. Coleman map
  12. Selmer groups and main conjectures
  13. Ordinary family
  14. Selmer groups and complexes
  15. Equivalence of main conjectures
  16. ...and 23 more sections

Key Result

Theorem 1.3

Assume $\mathbb{I}$ is Gorenstein. Suppose $\square$ and $\triangle$ are nearby (Definition def:Nearby), and $\triangle$ is incoherent, then there is a regulator map If $\mathrm{reg}(\boldsymbol{z}_\triangle)\neq 0$ for some $\boldsymbol{z}_\triangle\in\widetilde{\mathrm{H}}^1_\triangle(\mathcal{K},\boldsymbol{T})$, then the incoherent main conjecture for $\triangle$ and special cycle $\boldsymbo

Theorems & Definitions (37)

  • Conjecture 1.1: Conjecture \ref{['conj:MainConjecture']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['thm:Equiv']}
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Remark 3.1
  • Corollary 3.2
  • proof
  • ...and 27 more