Related papers: Un cas PEL de la conjecture de Kottwitz

We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As…

We express the cohomology of the basic stratum of some unitary Shimura varieties associated to division algebras in terms of automorphic representations of the group in the Shimura datum.

We introduce a ``hybrid'' conjecture which is a common generalisation of the Andr\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid…

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient…

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up…

In this article, we give a concrete description of the underlying reduced subscheme of the Rapoport--Zink spaces for spinor similitude groups with special maximal parahoric (and non-hyperspecial) level structure. Moreover, we give two…

We prove the Kudla-Rapoport conjecture for unramified unitary groups with maximal parahoric level structure. Our approach differs from the local proof given in Li-W.Zhang. We reduce the conjecture to a global intersection problem using…

We formulate a conjectural Lefschetz formula for locally symmetric spaces of finite volume. The formula can be verified in the compact case and for Riemann surfaces.

We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based…

Let $\kappa$ be a commutative ring containing $2^{-1}$. In this paper, we prove the Comes-Kujawa's conjecture on a $\kappa$-basis of cyclotomic oriented Brauer-Clifford supercategory. As a by-product, we prove that the cyclotomic walled…

The Test Function Conjecture due to Haines and Kottwitz predicts that the geometric Bernstein center is a source of test functions required by the Langlands-Kottwitz method for expressing the local semisimple Hasse-Weil zeta function of a…

We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and…

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz $p$-Curvature Conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties ${\cal A}_g$ when $g > 1.$

We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough…

We describe the structure of the supersingular Rapoport-Zink space associated to the group of unitary similitudes of signature (2,n-2) for an unramified quadratic extension of p-adic fields. In earlier work, two of the authors described the…

Let G be a unitary group over the rationals, associated to a CM-field F with totally real part F^+, with signature (1,1) at all the archimedean places of F^+. Under certain hypotheses on F^+, we show that Jacquet-Langlands correspondences…

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain…

Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the \'etale cohomology with…

Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions…

Using a mixed-characteristic incarnation of fusion, we prove an analog of Nekov\'a\v{r}-Scholl's plectic conjecture for local Shimura varieties. We apply this to obtain results on the plectic conjecture for (global) Shimura varieties after…