On Frobenius conjugacy class of some algebraic varieties over number fields

Abhijit Laskar (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let $X$ be a smooth projective algebraic variety over a number field $F$. The action of $ \mathrm{Gal}(\bar{F}/ F)$ on $\ell$-adic cohomology groups $ H^i_{et}(X_{/\bar{F}},\mathbb{Q}_{\ell})$, induces Galois representations $ \rho^i_{\ell}: \mathrm{Gal}(\bar{F}/ F) \rightarrow \mathrm{GL}(\mathrm{H}^i_{et}(X_{/\bar{F}},\mathbb{Q}_{\ell}))$. Fix a non-archimedean valuation $v$ on $F$. Let $F_v$ be the completion of $F$ at $v$, $\Phi_v$ be any arithmetic Frobenius element at $v$ and $W_v$ be the Weil group of $F_v$. First, we exhibit new cases of $X$ for which the characteristic polynomial of $\rho^i_{\ell}(w)$ has coefficients in $\mathbb{Q}$ and independent of $\ell$, for any $w \in W_v$. Then we establish new cases, for which the representations $({\rho^i}'_\ell, N'_{i,\ell}) $ of the Weil-Deligne group ${}'W_v$ of $F_v$, are Frobenius semisimple and form a compatible system of representations of ${}'W_v$, thus giving partial answer to a conjecture of Deligne, Tate, et al. Finally, we treat the question of $\ell$-independence of the conjugacy classes of $\rho^i_{\ell}(\Phi_v)$ in certain natural subgroups of $\mathrm{GL}(H^i_{et}(X_{/\bar{F}},\mathbb{Q}_{\ell}))$, both when $\ell$ is different or equal, to the residual characteristic at $v$.

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

Abhijit Laskar. On Frobenius conjugacy class of some algebraic varieties over number fields. 2013. ⟨hal-00804697v4⟩