Fix a prime number p [ges ] 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r [ges ] 1), which are denoted by ESp(N) ${\biib Z}$p and GES p(N)${\biib Z}$p, respectively. Let e* ′ be the projector to the direct sum of the ωi-eigenspaces of the ordinary part, for i [nequiv ] 0, −1 mod p−1. Our main result states that e* ′ GESp (N)${\biib Z}$p has a good p-adic Hodge structure, which can be described in terms of λ-adic modular forms, extending the previously known result for e*′ ESp (N)${\biib Z}$p. We then apply the method of Harder and Pink to the Galois representation on e*′ ESp(N) ${\biib Z}$p to construct large unramified abelian p-extensions over cyclotomic ${\bib Z}$p-extensions of abelian number fields.

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink‘s results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.

To a polynomial map f : Cn + 1 → C one can attach a monodromy transformation on the complex cohomology of its generic fiber that reflects its asymptotic behaviour. In this paper this transformation is determined for a class of generic polynomials in terms of data attached to projective hypersurfaces with isolated singularities.

The classical limit of the scaled elliptic algebra${\mathcal A}$ħ,η ($\widehat{\mathfrak sl}$2) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra ${\mathcal A}$q,p ($\widehat{\mathfrak sl}$2) is also briefly presented.

We constuct and investigate a structure of Verma-like modules over generalized Witt algebras. We also prove Futorny-like theorem for irreducible weight modlues whose dimensions of the weight spaces are uniformly bounded.

Let X be an m dimensional smooth projective variety with a Kähler metric. We construct a metrized line bundle $\cL$ with a rational section s over the product $\cC$p(X)× $\cC $q(X) of Chow varieties $\cC$p(X), $\cC$q(X) such that $\[{1\over (m-1)!}\log|s(A,B)|^2=\langle A, B\rangle \]$ for disjoint A, B. That gives an answer to a part of Barry Mazur‘s proposal in a private communication to Bruno Horris about the Archimedean height pairing 〈 A, B〉 on a smooth projective variety X.

Cet article constitue une partie importante de ma thèse. Je l‘ai écrit sous la direction avisée de Henri Carayol, à qui je dois un très grand merci car ce sont ses conseils et son optimisme qui m‘ont permis d‘en venir à bout. Je tiens également à remercier Laurent Clozel pour toutes ses remarques, utiles et pertinentes.

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.

Let A=(aij) be an orthogonal matrix (over R or C) with no entries zero. Let B= (bij) be the matrix defined by bij= 1/ai j. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon‘s theorem.

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of ’weights‘ (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.

We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ’hypergeometric sensitivity analysis‘ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.

We show that recently constructed invariants of 3-dimensional manifolds and of hyperkähler manifolds (L. Rozansky and E. witten) come from characteristic classes of foliations and from Gelfand-fuks cohomology.