Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Bloch-Kato conjecture for GSp(4)

David Loeffler, Sarah Livia Zerbes

Abstract

We prove an explicit reciprocity law for the Euler system attached to the spin motive of a genus 2 Siegel modular form. As consequences, we obtain one inclusion of the Iwasawa Main Conjecture for such motives, and the Bloch--Kato conjecture in analytic rank 0 for their critical twists.

On the Bloch-Kato conjecture for GSp(4)

Abstract

We prove an explicit reciprocity law for the Euler system attached to the spin motive of a genus 2 Siegel modular form. As consequences, we obtain one inclusion of the Iwasawa Main Conjecture for such motives, and the Bloch--Kato conjecture in analytic rank 0 for their critical twists.

Paper Structure

This paper contains 158 sections, 156 theorems, 411 equations, 2 figures.

Table of Contents

  1. Aims of this paper
  2. Main results of the paper
  3. Strategy
  4. Strategy for Theorem A
  5. Strategy for Theorem B
  6. Acknowledgements
  7. Euler systems for Siegel automorphic representations
  8. Automorphic representations
  9. Hecke parameters at $p$
  10. Shimura varieties
  11. Galois representations and Euler system classes
  12. Exponential maps and regulators
  13. Periods and $p$-adic $L$-functions
  14. Explicit formulation
  15. Finite-polynomial cohomology and Abel--Jacobi maps
  16. ...and 143 more sections

Key Result

Theorem A

Suppose $\Pi$ is unramified and Klingen-ordinary at $p$, and $r_1 - r_2 \geqslant 3$. Let $\nu$ be a basis of $\mathop{\mathrm{Gr}}\nolimits^1 \mathbf{D}_{\mathrm{dR}}(V_{\Pi})$, and let $\nu_{\mathrm{dR}}$ be its unique lifting to a vector in $\mathop{\mathrm{Fil}}\nolimits^1 \mathbf{D}_{\mathrm{dR for an explicit non-zero factor $(\star)$. Here, $\mathcal{L}_{p}(\Pi, \mathbf{j}_1, \mathbf{j}_2)$

Figures (2)

  • Figure 1: Visual representation of the strata in $Y_{\mathop{\mathrm{Kl}}\nolimits, 0}$. (Since the $\mathop{\mathrm{GSp}}\nolimits_4$ Shimura variety is 3-dimensional, and we are attempting to draw it on a 2-dimensional page, we have shrunk all the dimensions by one; hence the single point marked "$0, \alpha$" actually stands for a curve, etc.)
  • Figure 2: Intersections of boundary strata with EKOR strata.

Theorems & Definitions (413)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark 1
  • Remark 5.1.2
  • Remark 5.1.3
  • Definition 5.2.1
  • Remark 5.2.3
  • Definition 5.2.4
  • ...and 403 more