In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kähler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on ${\mathbb{P}}^2$ blown-up k points in general position, with $k<m+2$.

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic, then the $L_p$ compression of the wreath product $\mathbb{Z}\wr H$ equals ${\displaystyle \max \{\frac{1}{p},\frac{1}{2}\}}$. We also show that the $L_p$ compression of $\mathbb{Z}\wr \mathbb{Z}$ equals ${\displaystyle \max \{\frac{p}{2p-1},\frac{2}{3}\}}$ and that the $L_p$ compression of $(\mathbb{Z}\wr \mathbb{Z}{)}_0$ (the zero section of $\mathbb{Z}\wr \mathbb{Z}$, equipped with the metric induced from $\mathbb{Z}\wr \mathbb{Z}$) equals ${\displaystyle \max \{\frac{p+1}{2p},\frac{3}{4}\}}$. The fact that the Hilbert compression exponent of $\mathbb{Z}\wr \mathbb{Z}$ equals $2/3$ while the Hilbert compression exponent of $(\mathbb{Z}\wr \mathbb{Z}{)}_0$ equals $3/4$ is used to show that there exists a Lipschitz function $f:(\mathbb{Z}\wr \mathbb{Z}{)}_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\wr \mathbb{Z}$.

We introduce and study certain truncated sums of Hecke eigenvalues of ${\mathrm{GL}}_2$-automorphic forms along quadratic polynomials. A power-saving estimate is established, and new applications to moments of critical $L$-values associated to quadratic fields are derived. An application to the asymptotic behavior of the height of Heegner points and singular moduli is discussed in detail.

We study analytic continuation of overconvergent modular forms on Siegel varieties. We first analyze the dynamic of Hecke correspondances at $p$ over Siegel varieties with parahoric-level structure. We then concentrate on genus $2$ and prove a classicity criterion: a Siegel overconvergent modular form, of weight $(k_1,k_2)$, eigen for $U_p$ with eigenvalue $a_p$, such that $k_2>v(a_p)+3$ is classical. This implies that genus $2$ cuspidal ordinary $p$-adic modular forms of weight $(k_1,k_2)$ with $k_1\ge k_2\ge 4$ are classical.

Nous étudions la possibilité de réaliser le prolongement analytique des formes modulaires surconvergentes sur les variétés de Siegel. Dans un premier temps, nous analysons la dynamique des correspondances de Hecke en $p$ sur les variétés de Siegel avec niveau parahorique. Nous nous intéressons ensuite au genre $2$ et démontrons un critère de classicité: toute forme modulaire de Siegel surconvergente de poids $(k_1,k_2)$, propre pour $U_p$ pour la valeur propre $a_p$, avec $k_2>3+v(a_p)$ est classique. Ceci entraîne que les formes modulaires $p$-adiques ordinaires cuspidales de Siegel de genre $2$ et de poids $(k_1,k_2)$ avec $k_1\ge k_2\ge 4$ sont classiques.