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On Selmer complexes, Stark systems and derived $p$-adic heights

Daniel Macias Castillo, Takamichi Sano

Abstract

We develop the theory of Nekovář's Selmer complexes. We prove that, under mild hypotheses, Nekovář's Selmer complexes are canonically quasi-isomorphic to ``Poitou-Tate complexes", which arise from Poitou-Tate global duality exact sequences. We give two applications. Firstly, we prove that the determinant of a Selmer complex is canonically isomorphic to the module of Stark systems and, by using this result, we construct a canonical ``Heegner point Stark system" which controls Selmer groups. Secondly, we prove that the derived $p$-adic height pairing of Bertolini-Darmon concides with that of Nekovář.

On Selmer complexes, Stark systems and derived $p$-adic heights

Abstract

We develop the theory of Nekovář's Selmer complexes. We prove that, under mild hypotheses, Nekovář's Selmer complexes are canonically quasi-isomorphic to ``Poitou-Tate complexes", which arise from Poitou-Tate global duality exact sequences. We give two applications. Firstly, we prove that the determinant of a Selmer complex is canonically isomorphic to the module of Stark systems and, by using this result, we construct a canonical ``Heegner point Stark system" which controls Selmer groups. Secondly, we prove that the derived -adic height pairing of Bertolini-Darmon concides with that of Nekovář.
Paper Structure (34 sections, 35 theorems, 277 equations)

This paper contains 34 sections, 35 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Selmer and Poitou-Tate complexes
  4. Stark systems
  5. The general theory
  6. The Heegner point Stark system
  7. Derived $p$-adic heights
  8. Organization
  9. Notation
  10. Selmer complexes
  11. Nekovář's Selmer complexes
  12. Poitou-Tate complexes
  13. Proof of Theorem \ref{['qithm']}
  14. Preliminaries
  15. An explicit representative
  16. ...and 19 more sections

Key Result

Theorem 1.1

Under mild hypotheses, there exists a canonical quasi-isomorphism such that the following diagrams are commutative: \xymatrix{ H^1(C_{\rm Nek})/p^n \ar[dr]^{q_1} \ar[d]_{\kappa_1}& \\ H^1_\mathcal{F}(K,A)=H^1(C_{\rm PT}) \ar[r]_-{H^1(\varphi)} & H^1(C_{\rm Nek}/p^n), }\xymatrix{ H^2(C_{\rm Nek})/p^n \ar[rd]^{q_2} \ar[d]_{-\kappa_2}& \\ H^1_{\mathcal{F}^\ast}(K,A

Theorems & Definitions (86)

  • Theorem 1.1: See Theorem \ref{['qithm']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: See Theorem \ref{['thm stark']}
  • Remark 1.6
  • Theorem 1.7: See Theorem \ref{['thm heeg']}
  • Theorem 1.8: See Theorem \ref{['main']}
  • Example 2.1
  • Definition 2.2
  • ...and 76 more