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1 results

  • Platitude du pro-$p$-module universel de GL$_{2}(F)$ en caractéristique $p$

  • Rachel Ollivier
  • Journal:
    Compositio Mathematica / Volume 143 / Issue 3 / May 2007
    Published online by Cambridge University Press:
    04 December 2007, pp. 703-720
    Print publication:
    May 2007
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  • Let $F$ be a $p$-adic field with uniformizer $\pi$. We consider $G={\rm GL}_{2}(F)/\pi$ and $I(1)$ the pro-$p$-Iwahori subgroup of $G$. The exploration of the smooth mod $p$ representations of $G$ motivates the study of the space of functions with values in $\overline{\mathbb{F}}_p$ and compact support in the set of right cosets $I(1)\backslash G$. We show that this universal module is flat over the pro-$p$-Hecke algebra if and only if the cardinal of the residue field of $F$ is equal to $p$.