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Plectic structures in p-adic de Rham cohomology

David Loeffler, Sarah Livia Zerbes

Abstract

Given a Hilbert modular form for a totally real field $F$, and a prime $p$ split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius maps, indexed by the primes of $F$ above $p$. The general plectic conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism" comparing these structures to etale cohomology. We prove this conjecture in the case $[F : \mathbf{Q}] = 2$ under some mild assumptions; and for general $F$ we prove a weaker statement which is strong evidence for the conjecture, showing that plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii. (In memory of Jan Nekovar)

Plectic structures in p-adic de Rham cohomology

Abstract

Given a Hilbert modular form for a totally real field , and a prime split completely in , the -eigenspace in -adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius maps, indexed by the primes of above . The general plectic conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism" comparing these structures to etale cohomology. We prove this conjecture in the case under some mild assumptions; and for general we prove a weaker statement which is strong evidence for the conjecture, showing that plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii. (In memory of Jan Nekovar)
Paper Structure (22 sections, 19 theorems, 35 equations)

This paper contains 22 sections, 19 theorems, 35 equations.

Table of Contents

  1. Setup
  2. The spaces Dp(f) and Vp(f)
  3. Plectic structures
  4. Tensor induction
  5. Relation to the plectic conjectures
  6. Results for general d: statements
  7. The plectic filtration
  8. Preliminaries on I-filtrations
  9. BGG complexes
  10. Proof of \ref{['thm:main']}, I: geometric Jacquet-Langlands
  11. The S-ordinary locus
  12. The spectral sequence of Goren--Oort strata
  13. A new filtration
  14. Proof of \ref{['thm:main']}, II: higher Coleman theory
  15. Higher Coleman theory: statements
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $S \subseteq \{1, \dots, d\}$. For each $i \in S$, assume that the Hecke polynomial at $\mathfrak{p}_i$ has distinct roots, and that one of these roots $\alpha_i$ has strictly small slope. Then the simultaneous eigenspace $\bigcap_{i \in S} D_p(f)^{\varphi_i = \alpha_i}$ has dimension $2^{d - |S is an isomorphism. Moreover, this isomorphism is strictly compatible with the partial filtrations,

Theorems & Definitions (48)

  • Conjecture 1.1: Plectic comparison conjecture
  • Remark 1.2
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • ...and 38 more