Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex $\mathrm{UPic}(\overline{X})$. It is a complex of length 2 which combines the Picard group $\mathrm{Pic}(\overline{X})$ and the group $U(\overline{X}):=\overline{k}{[\overline{X}]}^{\times }/{\overline{k}}^{\times }$, where $\overline{k}$ is a fixed algebraic closure of k and $\overline{X}=X{\times }_k\overline{k}$. For a connected linear k-group G we compute the complex $\mathrm{UPic}(\overline{G})$ (up to a quasi-isomorphism) in terms of the algebraic fundamental group ${\pi }_1(\overline{G})$. We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Estimation du nombre de dérivées d'un processus Gaussien

We consider a real Gaussian process X with unknown smoothness $r_0\in \mathbb{N}$ where the mean-square derivative $X^{(r_0)}$ is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations $X(t_1),\dots ,X(t_n)$, we study reconstruction of $X(t)$, $t\in [0,1]$, with ${\stackrel{?}{X}}_r(t)$, a piecewise polynomial interpolation of degree $r?1$. We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as $r?r_0$. Next, from an interpolation-based empirical criterion, we derive an estimator $\stackrel{?}{r}$ of $r_0$ and prove its strong consistency by giving an exponential inequality for $P(\stackrel{?}{r}\ne r_0)$. Finally, we prove the strong convergence of ${\stackrel{?}{X}}_{\stackrel{?}{r}}(t)$ toward $X(t)$ with a similar rate as in the case ‘$r_0$ known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Modularity of hypertetrahedral representations

Let F be a number field, $G_F$ its absolute Galois group, and $\rho :G_F\to {\mathrm{GL}}_4(\mathbb{C})$ an irreducible continuous Galois representation. Let $\overline{G}$ denote the projective image of ρ in ${\mathrm{PGL}}_4(\mathbb{C})$. We say that ρ is hypertetrahedral if $\overline{G}$ is an extension of $A_4$ by the Klein group $V_4$. In this case, we show that ρ is modular, i.e., ρ corresponds to an automorphic representation π of ${\mathrm{GL}}_4({\mathbb{A}}_F)$ such that their L-functions are equal. This gives new examples of irreducible 4-dimensional monomial representations which are modular, but are not induced from normal extensions and are not essentially self-dual. To cite this article: K. Martin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).