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Euler systems for $\mathrm{GSp}(4)$ over imaginary quadratic fields

Alexandros Groutides

TL;DR

The paper constructs an Euler system attached to a general-type cohomological automorphic representation $\Pi$ of $\mathrm{GSp}_4/\mathbb{Q}$, twisted by a Groessencharacter $\psi$ of an imaginary quadratic field $K$, by adapting the $\mathrm{GSp}_4\times\mathrm{GL}_2$ classes of Hsu–Jin–Sakamoto via Gejima’s local input and Novodvorsky–Piatetski-Shapiro zeta integrals. It then transfers these motivic classes to Galois cohomology using an Abel–Jacobi map and CM-patching, producing Euler-system elements with explicit norm-relations over ray-class fields of $K$, and uses them to bound the strict Selmer group $\mathrm{Sel}_{\Sigma_p}(K,T^\vee(1))$ under standard hypotheses. The work also extends the $\mathrm{GSp}_4\times\mathrm{GL}_2$ Euler system to a motivic framework, covering certain small weights, and provides a coherent strategy to deduce Selmer bounds from Euler-system norm relations in a higher-rank setting. Overall, the results contribute a concrete Euler-system construction for $\mathrm{GSp}_4$ over imaginary quadratic fields and demonstrate its arithmetic applications via CM-patching and motivic norm-relations.

Abstract

We construct an Euler system attached to general-type cohomological cuspidal automorphic representations of $\mathrm{GSp}(4)$ twisted by a Groessencharacter of an imaginary quadratic field. We then use this to bound strict Selmer groups under standard hypotheses. In addition, our approach gives a way of extending the $\mathrm{GSp}(4)\times\mathrm{GL}(2)$ Euler system of Hsu-Jin-Sakamoto to a motivic statement which also covers certain small weights omitted in op$.$cit.

Euler systems for $\mathrm{GSp}(4)$ over imaginary quadratic fields

TL;DR

The paper constructs an Euler system attached to a general-type cohomological automorphic representation of , twisted by a Groessencharacter of an imaginary quadratic field , by adapting the classes of Hsu–Jin–Sakamoto via Gejima’s local input and Novodvorsky–Piatetski-Shapiro zeta integrals. It then transfers these motivic classes to Galois cohomology using an Abel–Jacobi map and CM-patching, producing Euler-system elements with explicit norm-relations over ray-class fields of , and uses them to bound the strict Selmer group under standard hypotheses. The work also extends the Euler system to a motivic framework, covering certain small weights, and provides a coherent strategy to deduce Selmer bounds from Euler-system norm relations in a higher-rank setting. Overall, the results contribute a concrete Euler-system construction for over imaginary quadratic fields and demonstrate its arithmetic applications via CM-patching and motivic norm-relations.

Abstract

We construct an Euler system attached to general-type cohomological cuspidal automorphic representations of twisted by a Groessencharacter of an imaginary quadratic field. We then use this to bound strict Selmer groups under standard hypotheses. In addition, our approach gives a way of extending the Euler system of Hsu-Jin-Sakamoto to a motivic statement which also covers certain small weights omitted in opcit.

Paper Structure

This paper contains 26 sections, 25 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Outline of the construction and main results
  3. Acknowledgments
  4. General notation
  5. Local structure theorems and zeta integrals
  6. Gejima's decomposition
  7. Passing to $\mathcal{I}(G(\mathbf{Q}_\ell)/G(\mathbf{Z}_\ell))$
  8. The principal-series and Siegel sections
  9. Novodvorsky’s zeta-integral
  10. Piatetski-Shapiro's zeta-integral and Bessel models
  11. Generic representations admit split Bessel models
  12. Relating Novdvorsky's and Piatetski-Shapiro's zeta integrals
  13. Motivic classes for $G$
  14. Input data
  15. Motivic $\mathop{\mathrm{GSp}}\nolimits_4\times \mathop{\mathrm{GL}}\nolimits_2$ tame norm-relations
  16. ...and 11 more sections

Key Result

Theorem A

Let $\mathscr{A}$ be the set of ideals of $\mathcal{O}_K$ whose norm is coprime to $cpN_\Pi N_{K/\mathbf{Q}}(\mathfrak{f})\Delta_K.$ Then for any $\mathfrak{n,l}\in\mathscr{A}$ with $\ell=\mathfrak{l\overline{l}}\nmid\mathfrak{n}$, a split prime, we have where $Q_\mathfrak{l}(X)=\det(1-X\ \mathrm{Frob}_\mathfrak{l}^{-1}|V_\Pi(\psi_\mathfrak{P})(k_2-1))$, $\mathrm{Frob}_\mathfrak{l}\in\mathrm{Gal}

Theorems & Definitions (72)

  • Remark 1.1.1
  • Theorem A: \ref{['thm split norm-relations in Galois']}
  • Remark 1.1.2
  • Theorem B: \ref{['thm selmer bound']}
  • Theorem 3.1.1: Gejima2018AnEF
  • proof
  • Lemma 3.1.2: Gejima2018AnEF
  • proof
  • Corollary 3.1.3
  • proof
  • ...and 62 more