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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...
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...
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....
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...
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...
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...
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...
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...
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