ABOUT GALOIS REDUCIBILITY OF TORSION COHOMOLOGY CLASSES FOR KHT SHIMURA VARIETIES

Pascal Boyer (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

We pursue the original strategy of my paper at JIMJ and we give a new criterion so that the localization of the cohomology of KHT Shimura variety is free. Precisely let $G$ be a similitude group with signatures $(1,d-1),(0,d),\cdots,(0,d)$, and $V_{\xi,\mathbb Z_l}$ a local system associated to a fixed algebraic representation $\xi$ of $G(\mathbb Q)$. Consider a system $\mathfrrak m$ of Hecke eigenvalues appearing in the free quotient of the cohomology group in middle degree of the Shimura variety $Sh_K$ associated to $G$ and with coefficients in $V_{\xi,\mathbb Z_l}$. Then if the modulo $l$ galoisian representation $\bar \rho_{\mathfrak m}$ is irreducible of dimension $d$, and essentially if $l \geq d+1$, the localization at $\mathfrak m$ of every cohomology group of $Sh_K$ with coefficients in $V_{\xi,\mathbb Z_l}$, is free.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Pascal Boyer. ABOUT GALOIS REDUCIBILITY OF TORSION COHOMOLOGY CLASSES FOR KHT SHIMURA VARIETIES. 2019. ⟨hal-02080476v1⟩