On Frobenius conjugacy class of some algebraic varieties over number fields

Abhijit Laskar (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let $X$ be smooth projective algebraic variety over a number field $F$ and $\bar{F}$ an algebraic closure of $F$. The action of $ \mathrm{Gal}(\bar{F}/ F)$ on the $\ell$-adic etale cohomology groups $ H^i_{et}(X_{/\bar{F}},\ql)$, induces Galois representations $ \rho^i_{\ell}: \mathrm{Gal}(\bar{F}/ F) \rightarrow \mathrm{GL}(H^i_{et}(X_{/\bar{F}},\ql))$. Fix a non-archimedean valuation $v$ on $F$ and let $\Phi_v $ be an arithmetic Frobenius element at $v$. In this article we answer, for many algebraic varieties, classical semisimplicty and $\ell$-independence questions related to $ \rho^i_{\ell}( \Phi_v)$ and its conjugacy class in certain natural subgroups of $\mathrm{GL}(H^i_{et}(X_{/\bar{F}},\ql))$. We deal with both the good reduction and bad reduction case. We also treat the case where $\ell$ is equal to the residual characteristic at $v$, by studying crystalline realization of motives.

Domaines: Théorie des nombres [math.NT]

Abhijit Laskar. On Frobenius conjugacy class of some algebraic varieties over number fields. 2013. ⟨hal-00804697v1⟩