Using non-abelian cohomology we introduce new obstructions to the Hasse principle. In particular, we generalize the classical descent formalism to principal homogeneous spaces under noncommutative algebraic groups and give explicit examples of application.

A uniform version of the Shafarevich Conjecture for function fields (Theorem of Parshin–Arakelov) is proved, together with a uniform version of the geometric Mordell conjecture (Theorem of Manin). Such results are generalized to base varieties of arbitrary dimension (i.e. function fields of arbitrary transcendence degree).

The Mellin transform of the fibre integral is calculated for certain quasihomogeneous isolated complete intersection singularities (above all, unimodal singularities of the list by Giusti and Wall). We show the symmetry property of the Gauss–Manin spectra (Theorem 3.1) and shed light on the lattice structure of the poles of the Mellin transform that are expressed by means of some topological data of the singularities (Theorem 4.3, Theorem 5.2). As an application of these results, we express the Hodge number of the fibre in terms of the Gauss–Manin spectra.

We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the finite analogs, using in part the p-adic version of polylogarithms and recent work of Besser. Kontsevich's original unpublished note is supplied (with his kind permission) in an ‘Appendix’ at the end of the paper.

We give an upper bound for the exponential sum [sum ]Mm=1 e2iπf(m) in terms of M and λ, where λ is a small positive number which denotes the size of the fourth derivative of the real valued function f. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G2(F5) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL2 to PGSp4.

T. Fukui introduced an invariant for the blow-analytic equivalence of real singularities. For a nondegenerate analytic function (germ) f, he discovered a formula for computing the one-dimensional invariant, denoted by A(f) := A1(f). We find a formula for A(f) for any f (real or complex, degenerate or not). We then define, and characterise, various notions of stability of A(f), using the formula. For real analytic f, the Fukui invariant with sign is defined, and computed by a similar formula. In the case where f is an analytic function of two complex variables, A(f) can also be computed using the tree-model of f.

The finite nth polylogarithm lin(z) ∈ Z/p(z) is defined as [sum ]k=1p−1zk/kn. We state and prove the following theorem. Let Lik: $ C$p → $ C$p be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1−nDFn(z) reduces modulo p>n+1 to lin−1(σ(z)), where D is the Cathelineau operator z(1−z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts [22, 23]). Standard Monomial Theory, developed by Seshadri and the first author [15, 16], and recently completed by the second author [20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of [20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in [12] (see also [13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar [10] and Mathieu [25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.

Let A be a local ring, and let I1,…,Ir ⊂ A be ideals of positive height. In this article we compare the Cohen–Macaulay property of the multi–Rees algebra RA(I1,…,Ir) to that of the usual Rees algebra RA(I1 ··· Ir) of the product I1 ··· Ir. In particular, when the analytic spread of I1 ··· Ir is small, this leads to necessary and sufficient conditions for the Cohen–Macaulayness of RA(I1,…,Ir). We apply our results to the theory of joint reductions and mixed multiplicities.

By using the idea of logarithm, we give an effective method to decide whether a stable series is an endomorphism of a formal group or not. We also give examples that contradict Lubin's conjecture.

We decompose the Marsden–Weinstein reductions for the moment map associated to representations of a quiver. The decomposition involves symmetric products of deformations of Kleinian singularities, as well as other terms. As a corollary we deduce that the Marsden–Weinstein reductions are irreducible varieties.

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.

Let k be a local field with residue characteristic p. Let n be a prime-to-p integer. Consider a smooth, irreducible k-curve X. In this article, we study the subgroup of H3(k(X), μn[otimes ]2) consisting of classes which are unramified on X. It is known to be isomorphic to H0(XZar, R3π*μn[otimes ]2), where π: Xét → XZar is the canonical map. A purely topological description of this group is given, using the Berkovich analytification Xan of X. More precisely, denote by Y a smooth and irreducible Berkovich-analytic k-curve, by Ytop the underlying topological space, by Yét the étale-analytic site and by πan the canonical map Yét → Ytop. Let Δ be the skeleton of Y (it is a closed subset defined by Berkovich which is locally a finite graph). Then we show (th. 42) that H0(Ytop, R3πan*μn[otimes ]2) is naturally isomorphic (through sort of a pointwise evaluation of cohomology classes) to the group of harmonic cochains defined on Δ with values in $Bbb Z$/n. Now, if X is a smooth algebraick-curve the natural map H0(XZar, R3π*μn[otimes ]2) → H0(Xantop, R3πan*μn[otimes ]2) is shown to be an isomorphism (th. 5.2). It is a new formulation (for the case where X is proper) and a generalization (to open curves) of a result which was due to Kato. Moreover we use here some steps of Kato's proof, but not his whole result. So our method gives, for projective curves, a (partially) new proof of Kato's theorem.