We generalize Colliot-Thélène’s construction of flasque resolutions of reductive group schemes over a field to a broad class of base schemes.

For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...

Le “principe de fonctorialité”, conjecturé par Langlands à la fin des années 60, est un moyen remarquablement synthétique d’unifier et exprimer certains liens profonds entre formes automorphes, arithmétique et géométrie algébrique. Son apparente simplicité contraste fortement avec la difficulté des techniques utilisées pour l’aborder. Parmi celles-ci, la stabilisation de la formule des traces d’Arthur–Selberg bute depuis 25 ans sur une conjecture d’analyse harmonique sur des groupes $p$-adiques :...

The Gauss-Schering Lemma is a classical formula for the Legendre symbol commonly used in elementary proofs of the quadratic reciprocity law. In this paper we show how the Gauss Schering Lemma may be generalized to give a formula for a $2$-cocycle corresponding to a higher metaplectic extension of GL${}_n/k$ for any global field $k$. In the case that $k$ has positive characteristic, our formula gives a complete construction of the metaplectic group and consequently an independent proof of the power reciprocity...

Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.