We formulate and prove necessary and sufficient conditions for the Galois correspondence to hold for the ring of p-adic periods B+dR.

Let X be an arbitrary variety over a finite field k and p=char k,n∈ N. We will construct a complex of étale sheaves on X together with trace isomorphism from the highest étale cohomology group of this complex onto Z/p$^n$Z such that for every constructible Z/p$^n$Z-sheaf on X the Yoneda pairing is a nondegenerate pairing of finite groups. If X is smooth, this complex is the Gersten resolution of the logarithmic de Rham–Witt sheaf introduced by Gros and Suwa. The proof is based on the special case proven by Milne when the sheaf is constant and X is smooth, as well as on a purity theorem which in turn follows from a theorem about the cohomological dimension of C$_i$-fields due to Kato and Kuzumaki. If the existence of the Lichtenbaum complex is proven, the theorem will be the p-part of a general duality theorem for varieties over finite fields.

In this paper, we develop a method of localization in equivariant cohomology based on the notion of partition of unity cohomology. We apply this method in two cases. In the first case, this method gives a refinement of the localization of Atiyah–Bott and Berline–Vergne (in the frame given by Bismut). After, we consider the Hamiltonian action of a torus, and we realise, following the idea of Witten, the localization on the critical points of the square of the moment map.

We give a combinatorial formula for the weight multiplicities of some infinite-dimensional highest weight ${\mathfrak g}$${\mathfrak l}$ (n)-modules. Our proof, which does not rely on Kazhdan–Lusztig combinatorics, uses a reduction to finite characteristics. The character formula for the corresponding modular representations, which has been computed in a 1997 preprint by the authors, is based on a dual pair which has no obvious counterpart in characteristic zero.

Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.

We associate to a group-like monoidal grupoid ${\mathcal C}$a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of${\mathcal C}$. The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.

Let Λ be a finite-dimensional hereditary algebra over a finite field k, ${\mathcal H}$(Λ) and ${\mathcal C}$(Λ) be, respectively, the Hall algebra and the composition algebra of Λ, ${\mathcal P}$ be the isomorphism classes of finite dimensional Λ-modules and I the isomorphism classes of simple Λ-modules. We define δ$_α$ and $_α$ δ, α in ${\mathcal P}$, to be the right and left derivations of ${\mathcal H}$(Λ) respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of Λ-mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u$_λ$ as the combinations of elements u$_i$ in the Hall algebra, where i ∈ I and λ in ${\mathcal P}$ is any exceptional Λ-module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.

Let D≡ 7 mod 8 be a positive squarefree integer, and let h$_D$ be the ideal class number of E$_D$=Q($\sqrt{−D}$). Let d≡1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[ges ]0 there is a constant M=M(k), independent of the pair (D,D), such that if (−1)$^k$=sign (d), (2k+1,h$_D$)=1, and$\sqrt{D}$>(12/π)d$^2$(log|d+M(k)), then the central L-value L(k+1, χ$^2k+1$$_D, d$>0. Furthermore, for k[les ]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.

Let k be a perfect field of characteristic p > 0, K$_0$ = Frac(W(k)), π a uniformizer in K$_0$ and π$_n$ ∈ K $_0$ (n∈ N) such that π$_0$ = π and π$^p$$_n$ $_ + 1$ = π$_n$. We write K$_∞$ = ∪$_n$ $_∈N$ K$_0$ (π$_n$), H$_∞$ = Gal (K$_0$/ K$_∞$ and G = Gal(K$_0$/ K$_0$). The main result of this paper is that the functor ’restriction of the Galois action‘ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length [les ] p − 2 to the category of p-adic representations of H$_∞$ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H$_∞$: one thanks to the norm field of K$_∞$, the other thanks to some categories of ’filtered‘ modules with divided powers previously introduced by the author.

Nambu structures are a generalization of Poisson structures in Hamiltonian dynamics, and it has been shown recently by several authors that, outside singular points, these structures are locally an exterior product of commuting vector fields. Nambu structures also give rise to co-Nambu differential forms, which are a natural generalization of integrable 1-forms to higher orders. This work is devoted to the study of Nambu tensors and co-Nambu forms near singular points. In particular, we give a classification of linear Nambu structures (integral finite-dimensional Nambu-Lie algebras), and a linearization of Nambu tensors and co-Nambu forms, under the nondegeneracy condition.

We establish sufficient conditions for a biextension algebra of a piecewise hereditary algebra of type Ā$_n$ or D$_n$ by indecomposable modules of derived regular length 2 to be of tame representation type.

We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements.

We introduce and describe the characteristic class of a difference operator over the difference field (k((t)),τ). Here k is an algebraically closed field of characteristic zero and τ is the k-linear automorphism of k((t)) defined by τ(t)=t/(1+t). The approach is based on the characterization of simple difference operators in terms of their eigenvalues.