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On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations

Naomi Sweeting

TL;DR

The paper advances the Bloch–Kato program for four-dimensional symplectic Galois representations V_{π,𝔭} attached to non-CAP, non-endoscopic automorphic π of GSp_4 with trivial central character and minimal archimedean weight. It develops a level-raising/bipartite Euler system framework using Kudla’s special cycles on ramified GSpin_5 Shimura varieties to produce ramified Galois cohomology classes, linking ramification to the central L-value L(π, spin, 1/2) via GSpin_3–GSpin_5 periods and theta lifts. In the rank-zero case, under mild hypotheses, nonvanishing L(π, spin, 1/2) forces H^1_f(ℚ, V_{π,𝔭}) to vanish (and the dual Selmer to vanish for almost all p), yielding evidence for Bloch–Kato in this setting; in the rank-one case, Kudla cycles with nontrivial Abel–Jacobi images imply dim_Ep H^1_f(ℚ, V_{π,𝔭}) = 1, with conjectural height formulas connected to L'(π, spin, 1/2) via an arithmetic Rallis inner product. The methods combine geometric input from Kudla cycles, automorphic theta lifts, weight spectral sequences, and deformation-theoretic level-raising to control congruences, addressing endoscopic phenomena and situating results within the broader framework of bipartite Euler systems and modularity-driven Galois realizations.

Abstract

The Bloch-Kato conjecture predicts a far-reaching connection between orders of vanishing of $L$-functions and the ranks of Selmer groups of $p$-adic Galois representations. In this article, we consider the four-dimensional, symplectic Galois representations arising from automorphic representations $π$ of $\mathrm{GSp}_4(\mathbb A_{\mathbb Q})$ with trivial central character and with the lowest cohomological archimedean weight. Under mild technical conditions, we prove that the Selmer group vanishes when the central value $L(π,\mathrm{spin},1/2)$ is nonzero. In the spirit of bipartite Euler systems, we bound the Selmer group by using level-raising congruences to construct ramified Galois cohomology classes. The relation to $L$-values comes via the $\mathrm{GSpin}_3\hookrightarrow \mathrm{GSpin}_5$ periods on a compact inner form of $\mathrm{GSp}_4$. We also prove a result towards the rank-one case: if the $π$-isotypic part of the Abel-Jacobi image of any of Kudla's one-cycles on the Siegel threefold is nonzero, it generates the full Selmer group. These cycles are linear combinations of embedded quaternionic Shimura curves, and under the conjectural arithmetic Rallis inner product formula, their heights are related to $L'(π,\mathrm{spin},1/2)$.

On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations

TL;DR

The paper advances the Bloch–Kato program for four-dimensional symplectic Galois representations V_{π,𝔭} attached to non-CAP, non-endoscopic automorphic π of GSp_4 with trivial central character and minimal archimedean weight. It develops a level-raising/bipartite Euler system framework using Kudla’s special cycles on ramified GSpin_5 Shimura varieties to produce ramified Galois cohomology classes, linking ramification to the central L-value L(π, spin, 1/2) via GSpin_3–GSpin_5 periods and theta lifts. In the rank-zero case, under mild hypotheses, nonvanishing L(π, spin, 1/2) forces H^1_f(ℚ, V_{π,𝔭}) to vanish (and the dual Selmer to vanish for almost all p), yielding evidence for Bloch–Kato in this setting; in the rank-one case, Kudla cycles with nontrivial Abel–Jacobi images imply dim_Ep H^1_f(ℚ, V_{π,𝔭}) = 1, with conjectural height formulas connected to L'(π, spin, 1/2) via an arithmetic Rallis inner product. The methods combine geometric input from Kudla cycles, automorphic theta lifts, weight spectral sequences, and deformation-theoretic level-raising to control congruences, addressing endoscopic phenomena and situating results within the broader framework of bipartite Euler systems and modularity-driven Galois realizations.

Abstract

The Bloch-Kato conjecture predicts a far-reaching connection between orders of vanishing of -functions and the ranks of Selmer groups of -adic Galois representations. In this article, we consider the four-dimensional, symplectic Galois representations arising from automorphic representations of with trivial central character and with the lowest cohomological archimedean weight. Under mild technical conditions, we prove that the Selmer group vanishes when the central value is nonzero. In the spirit of bipartite Euler systems, we bound the Selmer group by using level-raising congruences to construct ramified Galois cohomology classes. The relation to -values comes via the periods on a compact inner form of . We also prove a result towards the rank-one case: if the -isotypic part of the Abel-Jacobi image of any of Kudla's one-cycles on the Siegel threefold is nonzero, it generates the full Selmer group. These cycles are linear combinations of embedded quaternionic Shimura curves, and under the conjectural arithmetic Rallis inner product formula, their heights are related to .

Paper Structure

This paper contains 113 sections, 232 theorems, 925 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Overview of the proofs
  4. Sketch of the proof in the rank zero case
  5. Ramification and the relation to $L$-values
  6. Level-raising
  7. Sketch of the proof in the rank one case
  8. Comparison to other work
  9. Comments on the endoscopic case
  10. Organization of the paper
  11. Acknowledgements
  12. Preliminaries
  13. Notation
  14. Number fields and Galois groups
  15. Adèle groups and class field theory
  16. ...and 98 more sections

Key Result

Theorem 1

Suppose $\pi$ is not CAP or endoscopic, and for some $\ell|N(\pi)$, $\pi_\ell$ has a local Jacquet-Langlands transfer to the compact inner form of $\mathop{\mathrm{GSp}}\nolimits_{4,\mathbb{Q}_\ell}$. Let $\mathfrak{p}|p$ be a prime of $E$ such that: Then

Theorems & Definitions (567)

  • Theorem 1: Theorem \ref{['thm:rk_zero_main']}
  • Corollary 2: Corollary \ref{['cor:rk_zero_integral_IIa']}
  • Theorem 3: Theorem \ref{['thm:rk1_non_endoscopic_ultimate']}
  • Proposition 4: Theorem \ref{['thm:endoscopic_rk1']}
  • Proposition 1.2.2
  • proof
  • Proposition 1.2.4
  • proof
  • Lemma 1.2.6
  • proof
  • ...and 557 more