Considering a holonomic ${\mathcal D}$-module and a hypersurface, we define a finite family of ${\mathcal D}$-modules on the hypersurface which we call modules of vanishing cycles. The first one had been previously defined and corresponds to formal solutions. The last one corresponds, via Riemann-Hilbert, to the geometric vanishing cycles of Grothendieck-Deligne. For regular holonomic ${\mathcal D}$-modules there is only one sheaf and for non regular modules the sheaves of vanishing cycles control the growth and the index of solutions. Our results extend to non holonomic modules under some hypothesis.

Classifying spaces and moduli spaces are constructed for two invariants of isolated hypersurface singularities, for the polarized mixed Hodge structure on the middle cohomology of the Milnor fibre, and for the Brieskorn lattice as a subspace of the Gauß–Manin connection. The relations between them, period mappings for μ-constant families of singularities, and Torelli theorems are discussed.

We show that the Kodaira dimension of the moduli space of polarized K3 surfaces of degree 2n in non negative if n = 42, 43, 51, 53, 55, 57, 59, 61, 66, 67, 69, 74, 83, 85, 105, 119 or 133. We use an automorphic form associated with the fake monster Lie algebra constructed by Borcherds.

We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:

Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension [ges ] 2 or Gorenstein of dimension =1.

In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.

Every derivation of the Lie algebra is inner.

In this paper, we prove an analogue of the result known as Mazur‘s Principle concerning optimal levels of mod [ell ] Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of the special fibre. It is this study which allows us to generalise, in the second part of this paper, Mazur‘s result to totally real fields of odd degree.

We give a generic estimation of trigonometric sums defined over closed sub-schemes with semi-stable reduction of the standard affine scheme modulo pn(n [ges ] 2). We use Greenberg realisation to reduce to trigonometric sums defined over smooths sub-schemes of a finite product of Witt vectors over the finite field of p elements. Using the cohomological interpretation of this sums over a finite field, the sum is directly related to the Fourier–Deligne transformation of the dual pairs of Witt vectors. We deduce the estimation from the properties of the Fourier–Deligne transformation on simple perverse sheaves and pure sheaves.

We show that unipotent overconvergent isocrystals are algebraic and that the category of unipotent overconvergent isocrystals has a Frobenius automorphism. We also prove a structure theorem for unipotent overconvergent F-isocrystals over an open subset of the line, analogous to Dieudonné–Manin decomposition theorem for F-isocrystals.

We prove that a refinement of Stark‘s Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin‘s holds true.

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier‘s theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.

It is shown that if the fundamental group of a normal algebraic variety, respectively Zariski open subset of a compact Kähler manifold, is solvable with a faithful linear representation over Q, respectively polycyclic, then it is virtually nilpotent.

In this paper we study – for a semistable scheme – a comparison map between its log-syntomic cohomology and the p-adic étale cohomology of its generic fiber. The image can be described in terms of what Bloch and Kato call the local points of the underlying motive. The results extend a proven conjecture of Schneider which treats the good reduction case. The proof uses the theory of logarithmic schemes, some crystalline cohomology theories defined on them and various techniques in p-adic Hodge theory, in particular the Fontaine–Jannsen conjecture proven by Kato and Tsuji as well as Fontaine‘s rings of p-adic periods and their properties. The comparison result may become useful with respect to cycle class maps.

This paper addresses questions involving the sharpness of Vojta‘s conjecture and Vojta‘s inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta‘s inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta‘s conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojta‘s conjecture is best possible with D=0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta‘s conjecture is sharp with D=0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.