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Euler systems for conjugate-symplectic motives

Daniel Disegni

Abstract

Let $ρ$ be a conjugate-symplectic, geometric representation of the Galois group of a CM field. Under the assumption that $ρ$ is automorphic, even-dimensional, and of minimal regular Hodge--Tate type, we construct an Euler system for $ρ$ in the sense of forthcoming work of Jetchev--Nekovar--Skinner. The construction is based on Theta cycles as introduced in a previous paper, following works of Kudla and Liu on arithmetic theta series on unitary Shimura varieties; it relies on a certain modularity hypothesis for those theta series. Under some ordinariness assumptions, one can attach to $ρ$ a p-adic L-function. By recent results of Liu and the author, and the theory of Jetchev--Nekovar--Skinner, we deduce the following (unconditional) result under mild assumptions: if the p-adic L-function of $ρ$ vanishes to order 1 at the centre, then the Selmer group of $ρ$ has rank 1, generated by the class of an algebraic cycle. This confirms a case of the p-adic Beilinson--Bloch--Kato conjecture.

Euler systems for conjugate-symplectic motives

Abstract

Let be a conjugate-symplectic, geometric representation of the Galois group of a CM field. Under the assumption that is automorphic, even-dimensional, and of minimal regular Hodge--Tate type, we construct an Euler system for in the sense of forthcoming work of Jetchev--Nekovar--Skinner. The construction is based on Theta cycles as introduced in a previous paper, following works of Kudla and Liu on arithmetic theta series on unitary Shimura varieties; it relies on a certain modularity hypothesis for those theta series. Under some ordinariness assumptions, one can attach to a p-adic L-function. By recent results of Liu and the author, and the theory of Jetchev--Nekovar--Skinner, we deduce the following (unconditional) result under mild assumptions: if the p-adic L-function of vanishes to order 1 at the centre, then the Selmer group of has rank 1, generated by the class of an algebraic cycle. This confirms a case of the p-adic Beilinson--Bloch--Kato conjecture.

Paper Structure

This paper contains 42 sections, 2 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Idea of the proof
  4. The Euler system of Theta cycles
  5. Setup
  6. Notation
  7. Quasisplit unitary group
  8. Incoherent unitary groups
  9. Weil representation
  10. $p$-adic automorphic representations
  11. Realisation of $\sigma$ in cohomology
  12. Connected components of unitary Shimura varieties
  13. Special cycles
  14. Projection to the $\rho$-component
  15. Generating series
  16. ...and 27 more sections

Key Result

Theorem 1

Let $\rho$ be a representation satisfying conditions 1., 2., 3. above, and let $\mathscr{M}_{1}, \mathscr{M}, \wp, \mathscr{M}[\wp]$ be as above. Assume that the root number $\varepsilon(\rho)=-1$, that $F\neq\mathbf{Q}$ or $n=2$, and that the Modularity Hypothesis mod holds. The system of classes of Definition def th c forms a JNS Euler system.

Theorems & Definitions (30)

  • Theorem 1
  • Corollary
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem \ref{['norm rel']}.1, assuming Proposition \ref{['int']}
  • proof : Proof of Theorem \ref{['norm rel']}.\ref{['NRh']}, assuming Proposition \ref{["norm rel'"]}
  • proof : Proof of Proposition \ref{["norm rel'"]}, assuming Proposition \ref{['norm rel"']}
  • ...and 20 more