On admissible tensor products in $p$-adic Hodge theory

Abstract

We prove that if $W$ and $W^{\prime }$ are two $B$-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge-Tate), then there exists a character $\mu$ such that $W({\mu }^{-1})$ and $W^{\prime }(\mu )$ are crystalline (or semi-stable or de Rham or Hodge-Tate). We also prove that if $W$ is a $B$-pair and $F$ is a Schur functor (for example $\Sym^n(-)$ or ${\Lambda }^n(-)$) such that $F(W)$ is crystalline (or semi-stable or de Rham or Hodge-Tate) and if the rank of $W$ is sufficiently large, then there is a character $\mu$ such that $W({\mu }^{-1})$ is crystalline (or semi-stable or de Rham or Hodge-Tate). In particular, these results apply to $p$-adic representations.

References in corpus (1)

• Classification of two dimensional split trianguline representations of $p$-adic fields

Cited by in corpus (2)

• Geometrically irreducible $p$-adic local systems are de Rham up to a twist
• Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms