Let $\chi$ be a nontrivial multiplicative character of ${\mathbb{F}}_{p^n}$. We obtain the following results.

(1) Let $ϵ>0$ be given. If $B=\{\sum _{j=1}^n{x}_j{\omega }_j :x_j\in [N_j+1,N_j+H_j]\cap \mathbb{Z},j=1,\dots ,n\}$ is a box satisfying ${\Pi }_{j=1}^n{H}_j>p^{(2/5+ϵ)n},$ then for $p>p(ϵ)$ we have, denoting $\chi$ a nontrivial multiplicative character, ${\displaystyle |\sum _{x\in B}\chi (x)|{\ll }_n{p}^{-{ϵ}^2/4}\left|B\right|}$ unless $n$ is even, $\chi$ is principal on a subfield $F_2$ of size $p^{n/2}$, and ${\max }_{\xi }|B\cap \xi F_2|>p^{-ϵ}\left|B\right|$.

(2) Assume that $A,B\subset {\mathbb{F}}_p$ so that ${\displaystyle \left|A\right|>p^{(4/9)+ϵ},\left|B\right|>p^{(4/9)+ϵ},|B+B|<K\left|B\right|.}$ Then ${\displaystyle |\sum _{x\in A,y\in B}\chi (x+y)|<p^{-\tau }\left|A\right| \left|B\right|.}$

(3) Let $I\subset {\mathbb{F}}_p$ be an interval with $\left|I\right|=p^{\beta }$, and let $D\subset {\mathbb{F}}_p$ be a $p^{\beta }$-spaced set with $\left|D\right|=p^{\sigma }$. Assume that $2\beta +\sigma -\beta \sigma /(1-\beta )>1/2+\delta$. Then for a nonprincipal multiplicative character $\chi$, ${\displaystyle |\sum _{x\in I,y\in D}\chi (x+y)|<p^{-{\delta }^2/12}\left|I\right|\left|D\right|.}$ We also slightly improve a result of Karacuba [K3]

En suivant les travaux de Goresky, Kottwitz et MacPherson [12], on calcule l'homologie des fibres de Springer affines tronquées dans le cas non ramifié mais sous une hypothèse de pureté. Dans le cas équivalué, on démontre cette hypothèse de pureté. La troncature dépend d'un paramètre qui est un diviseur sur une variété torique $\ell$-adique. Dans les cas non ramifiés et équivalués, pour chaque fibre de Springer affine tronquée, on définit un faisceau cohérent gradué sur cette variété torique dont l'espace des sections globales s'identifie à l'homologie $\ell$-adique de la fibre considérée. Pour certaines familles de groupes endoscopiques, ces faisceaux apparaissent dans des suites exactes. On en déduit alors le lemme fondamental pondéré d'Arthur [5, conjecture 5.1] dans les cas équivalués et non ramifiés.

Following Goresky, Kottwitz, and MacPherson [12], we compute the homology of truncated affine Springer fibers in the unramified case but under a purity assumption. We prove this assumption in the equivalued case. The truncation parameter is viewed as a divisor on an $\ell$-adic toric variety. In the unramified and equivalued cases, for each truncated affine Springer fiber, we introduce a graded coherent sheaf on the toric variety whose space of global sections is precisely the $\ell$-adic homology of this fiber. Moreover, for some families of endoscopic groups, these sheaves show up in an exact sequence. As a consequence, we prove Arthur's weighted fundamental lemma [5, Conjecture 5.1] in the unramified equivalued case

For Schrödinger maps from ${\mathbb{R}}^2\times {\mathbb{R}}^{+}$ to the $2$-sphere ${\mathbb{S}}^2$, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space--dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map

Let $(M,g_0)$ be a compact Riemannian manifold of dimension $n\ge 4$. We show that the normalized Ricci flow deforms $g_0$ to a constant curvature metric, provided that $(M,g_0)\times \mathbb{R}$ has positive isotropic curvature. This condition is stronger than two-positive flag curvature but weaker than two-positive curvature operator