Let $F$ be a $p$-adic field, let $L$ be the completion of a maximal unramified extension of $F$, and let $\sigma$ be the Frobenius automorphism of $L$ over $F$. For any connected reductive group $G$ over $F$ one denotes by $B(G)$ the set of $\sigma$-conjugacy classes in $G(L)$ (elements $x,y$ in $G(L)$ are said to be $\sigma$-conjugate if there exists $g$ in $G(L)$ such that $g^-1 \kappa \sigma(g)=y$. One of the main results of this paper is a concrete description of the set $B(G)$ (previously this was known only in the quasi-split case).

We investigate the relationship between the usual and general Hodge conjectures for abelian varieties. For certain abelian varieties $A$, we show that the usual Hodge conjecture for all powers of $A$ implies the general Hodge conjecture for $A$.

Let $G$ a compact group of isometries of a closed riemannian manifold $(X,m)$. Sunada proved that if $\Gamma_1$ and $\Gamma_2$ are two finite almost-conjugated subgroups of $G$, then $\Gamma_1\setminus X,m)$ and $\Gamma_2\setminus X,m)$ are isospectral. We prove that if $G$ is finite, there exists an open dense set in the set of $G$-invariant metrics for which the converse of this resukt is true. If $G$ is infinite, the situations is more complicated and we obtain some partial results.