Volume 2006, Issue 591

Fouvry and Iwaniec's theorem concerning three-dimensional exponential sums with monomials relies on a spacing lemma whose optimal form is yet unproved. We bypass their spacing lemma via a diophantine problem in four variables and we obtain the expected bound in their theorem. In the problem of abelian groups of a given order, this yields the exponent 1/4 + ε, a result close to a conjecture of H. E. Richert (1952).

We define and investigate a Fourier-Mukai transformation for Higgs bundles on smooth curves. It can be viewed as the purely algebro-geometric or holomorphic side of a Nahm-type transformation for Higgs bundles. The transform of a stable degree-0 Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve; we show that it admits a natural extension to an algebraic vector bundle over a projective compactification of the base. The main result is that the original Higgs bundle can be recovered from this extension.

The inequality of Alexandrov, Bakel'man and Pucci is a basic tool in the theory of linear elliptic partial differential equations (PDEs) which are not in divergence form as well as in the more general theory of nonlinear elliptic PDEs. Here, in two dimensions, we prove the sharp form of the maximum principle as conjectured by Pucci in 1966, give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting. These results make use of the substantial recent advances in the planar theory of quasiconformal mappings.

Exploiting a notion of Kähler structure on a stratified space introduced elsewhere we show that, in the Kähler case, reduction after quantization is equivalent to quantization after reduction in a certain weak sense. Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kähler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization is weakly equivalent to quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond as complex vector spaces (whether or not the natural inner products correspond is unknown whence the qualifier “weakly”) but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kähler spaces arise from representations of compact Lie groups and from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces. For illustration, we carry out Kähler quantization on various spaces of that kind including singular Fock spaces.

The goal of this paper is to derive a number theoretic expression for the trace tr λ T v of the Hecke operator T v acting on the eigenspace ℰ λ for the eigenvalue λ = 1 + κ 2 > 0 of −Δ. We obtain that the trace can be expressed as the residue of a certain linear combination of L -series. Furthermore, the analytic behaviour of this combination gives information on the asymptotic behaviour of the class number.

The exceptional zero conjecture relates the first derivative of the p -adic L -function of a rational elliptic curve with split multiplicative reduction at p to its complex L -function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field 𝔽 q ( T ) with split multiplicative reduction at two places 𝔭 and ∞, avoiding the construction of a 𝔭-adic L -function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.

We consider Fano manifolds M that admit a collection of finite automorphism groups G 1 , …, G k , such that the quotients M / G i are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kähler-Einstein metric too.

We prove that ribbons, i.e. double structures associated with a line bundle ℰ over its reduced support, a smooth irreducible projective curve of arbitrary genus, are smoothable if their arithmetic genus is greater than or equal to 3 and the support curve possesses a smooth irreducible double cover with trace zero module ℰ. The method we use is based on the infinitesimal techniques that we develop to show that if the support curve admits such a double cover then every embedded ribbon over the curve is “infinitesimally smoothable”, i.e. the ribbon can be obtained as central fiber of the image of some first-order infinitesimal deformation of the map obtained by composing the double cover with the embedding of the reduced support in the ambient projective space containing the ribbon. We also obtain embeddings in the same projective space for all ribbons associated with ℰ. Then, assuming the existence of the double cover, we prove that the “infinitesimal smoothing” can be extended to a global embedded smoothing for embedded ribbons of arithmetic genus greater than or equal to 3. As a consequence we obtain the smoothing results.