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On local characterizations of Hida families of Siegel modular forms

Shaunak V. Deo, Bharathwaj Palvannan

Abstract

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from automorphic inductions (stable Yoshida lifts), analogous to the characterizations of Hida families of CM modular forms provided by Ghate--Vatsal. Our characterizations involve (i) density of de Rham at $p$ specializations at the singular weights $(k,2)$ and (ii) local decomposability at $p$ of the associated $Λ$-adic Galois representation. Our approach is similar to that of Castella--Wang-Erickson who provided an alternate strategy to reproving the main results of Ghate--Vatsal by applying Ribet's method when an anti-cyclotomic class group is assumed to be pseudo-null and cyclic as a $Λ$-module. Along these lines, one key input to our methods involves an assumption of pseudo-nullity of Selmer groups that are defined by imposing stricter conditions at $p$ than those imposed for the usual Greenberg Selmer groups appearing in the Asai main conjectures over real quadratic fields. Following Genestier--Tilouine and Pilloni, we also prove a minimal $R=\mathbb{T}$ theorem that is essential to establishing our results at various stages.

On local characterizations of Hida families of Siegel modular forms

Abstract

We provide new local characterizations of Hida families of Siegel modular forms with genus two arising from automorphic inductions (stable Yoshida lifts), analogous to the characterizations of Hida families of CM modular forms provided by Ghate--Vatsal. Our characterizations involve (i) density of de Rham at specializations at the singular weights and (ii) local decomposability at of the associated -adic Galois representation. Our approach is similar to that of Castella--Wang-Erickson who provided an alternate strategy to reproving the main results of Ghate--Vatsal by applying Ribet's method when an anti-cyclotomic class group is assumed to be pseudo-null and cyclic as a -module. Along these lines, one key input to our methods involves an assumption of pseudo-nullity of Selmer groups that are defined by imposing stricter conditions at than those imposed for the usual Greenberg Selmer groups appearing in the Asai main conjectures over real quadratic fields. Following Genestier--Tilouine and Pilloni, we also prove a minimal theorem that is essential to establishing our results at various stages.
Paper Structure (26 sections, 32 theorems, 222 equations)

This paper contains 26 sections, 32 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting: $p$-adic families of nearly ordinary Hilbert modular forms over real quadratic fields
  4. Setting: Yoshida lifts and Hida theory for $\mathrm{GSp}_4$
  5. Main Results
  6. Concluding Remarks
  7. Acknowledgements
  8. Hilbert modular forms over the real quadratic field $K$
  9. Hecke eigensystems associated to nearly ordinary Hilbert cuspidal eigenforms
  10. Hida theory for $\mathrm{GL}_2$ over $K$
  11. Siegel modular forms over $\mathbb{Q}$ with genus two
  12. Inductions of two dimensional representations
  13. Stable Yoshida lifts
  14. Hida's $p$-ordinary Hecke algebra for $\mathrm{GSp}_4$
  15. Shapes and properties of Galois representations
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that the following hypotheses hold for the residual Galois representation $\overline{\rho}_F$ associated to the $\Lambda$-adic Hida family $F$: Then, the following statements are equivalent.

Theorems & Definitions (82)

  • Theorem : Ghate--Vatsal MR2139691
  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Theorem 3
  • Remark 1.4
  • Theorem 4
  • Remark 1.5
  • ...and 72 more