Related papers: The Jacquet-Langlands correspondence for overconve…
We find explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper we proved that such formulas should exist. We give applications to the…
We consider a moduli space of lattice polarized K3 surfaces with the additional information of a frame of the trascendental cohomology with respect to the lattice polarization. This moduli space is proved to be quasi-affine, and the…
We generalize a result of Ribet and Takahashi on the parametrization of elliptic curves by Shimura curves to the Hilbert modular setting. In particular, we study the behaviour of the parametrization of modular abelian varieties by Shimura…
We construct big generalized Heegner classes by interpolating $p$-adically the generalized Heegner classes associated to quaternionic modular forms along a Coleman (finite slope) family, following the approach introduced by…
We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We…
Let $F$ be a locally compact non-Archimedean field, and let $B/F$ be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations of $B^\times$ of dimension $>1$ and…
This is the second article in a two-part series presenting a new proof comparing the non-invariant trace formula for a general linear group with that of one of its inner forms. In this article, we focus on the spectral side of the trace…
We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications…
The mod $p$ Riemann-Hilbert correspondence (in covariant and contravariant forms) relates $\mathbb{F}_p$-\'etale sheaves on the spectrum of an $\mathbb{F}_p$-algebra $R$ and Frobenius modules over $R$. We give an exposition of these…
For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…
We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…
In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this…
We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case.
We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study…
The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen. The aim of this article is to write translation for curves over the complex field and prove the correspondence in…
We give a classification of Jordan-Chevalley decompositions of an endomorphism of a finite-dimensional vector space over a not necessarily perfect field, i.e. additive decompositions into commuting semisimple and nilpotent endomorphisms.
The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that…
Infinite-dimensional manifolds modelled on arbitrary Hilbert spaces of functions are considered. It is shown that changes in model rather than changes of charts within the same model make coordinate formalisms on finite and…
We describe the algebra of invariants of the vacuum module associated with the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. We give a formula for its Hilbert--Poincar\'{e} series in a fermionic (cancellation-free) form which…
This is a survey of the proof of the global Jacquet-Langlands correspondence between GL_n and general inner forms, in characteristic zero. The proof is given after a long introduction recalling in some detail the objects and results…