Lifting Theorem Research Articles

Modulo ℓ distinction problems

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$ . This paper concerns the $\ell$ -modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$ . We start by proving a general theorem allowing to lift supercuspidal $\overline {\mathbf {F}}_{\ell }$ -representations of $\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$ -representation. Given a quadratic field extension $E/F$ and an irreducible $\overline {\mathbf {F}}_{\ell }$ -representation $\pi$ of $\operatorname {GL}_n(E)$ , we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi _\pi$ is irreducible and conjugate-selfdual, then $\pi$ is either $\operatorname {GL}_n(F)$ -distinguished or $(\operatorname {GL}_{n}(F),\omega _{E/F})$ -distinguished (where $\omega _{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$ . We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$ . After that, we give a complete classification of the $\operatorname {GL}_2(F)$ -distinguished representations of $\operatorname {GL}_2(E)$ . Using this classification we discuss a modular version of the Prasad conjecture for $\operatorname {PGL}_2$ . We show that the ‘classical’ Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil–Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\operatorname {SL}_2(F)$ -distinguished modular representations of $\operatorname {SL}_2(E)$ .

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Tropical extensions and Baker-Lorscheid multiplicities for idylls

In a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property.

A lifting theorem for planar mixed automorphic functions

A lifting theorem for planar mixed automorphic functions

Asymptotic freeness in tracial ultraproducts

Abstract We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$ , $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$ . Our proof relies on Mei–Ricard’s results [MR16] regarding $\operatorname {L}^p$ -boundedness (for all $1 < p < +\infty $ ) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan–Ioana–Kunnawalkam Elayavalli’s recent construction [CIKE22] to provide the first example of a $\mathrm {II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

Potential automorphy over CM fields

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbb{A}_F)$.

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue differentiation theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon–Nikodým property.

Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

We present some lifting theorems for continuous order-preserving functions on locally and σ-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and σ-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions.

On the modularity of elliptic curves over the cyclotomic $${\mathbb {Z}}_p$$-extension of some real quadratic fields

The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime p, every elliptic curve defined over a p-cyclotomic extension of \( {\mathbb {Q}} \) is modular. The method is to use some automorphy lifting theorems and study non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic curves. Since the modularity of elliptic curves over real quadratic was proved, one may ask whether it is possible to replace \({\mathbb {Q}}\) with a real quadratic field K. Following Thorne’s idea, we give some assumptions first and prove the modularity of elliptic curves over the \({\mathbb {Z}}_p\)-extension of some real quadratic fields.

Automorphy lifting with adequate image

Abstract Let F be a CM number field. We generalise existing automorphy lifting theorems for regular residually irreducible p-adic Galois representations over F by relaxing the big image assumption on the residual representation.

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $\rho : W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.

Modularity lifting theorems

Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.

Toeplitz operators and Hilbert modules on the symmetrized polydisc

When is the collection of [Formula: see text]-Toeplitz operators with respect to a tuple of commuting bounded operators [Formula: see text], which has the symmetrized polydisc as a spectral set, nontrivial? The answer is in terms of powers of [Formula: see text] as well as in terms of a unitary extension. En route, the Brown–Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the [Formula: see text]-algebra generated by the commutant of [Formula: see text] and the commutant of its unitary extension [Formula: see text].

Potential automorphy for $$GL_n$$

We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in \cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain $p$-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in \cite{tap}.

A note on Galois representations valued in reductive groups with open image

Let G be a split reductive group with dim⁡Z(G)≤1. We show that for any prime p that is large enough relative to G, there is a finitely ramified Galois representation ρ:ΓQ→G(Zp) with open image. We also show that for any given integer e, if the index of irregularity of p is at most e and if p is large enough relative to G and e, then there is a Galois representation ρ:ΓQ→G(Zp) ramified only at p with open image, generalizing a theorem of Ray [8]. The first type of Galois representation is constructed by lifting a suitable Galois representation into G(Fp) using a lifting theorem of Fakhruddin–Khare–Patrikis [4], and the second type of Galois representation is constructed using a variant of the argument in Ray's work [8].

Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods

Let e≥2 be an integer, pr be a prime power with pr≡1(mode) and ηr(i) be Gaussian periods of degree e for Fpr. By the dual form of Davenport and Hasse's lifting theorem on Gauss sums, we establish lifts of the multiplication matrices of the Gaussian periods ηr(0),…,ηr(e−1) which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree e with 3≤e≤23 which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.

Lifting theorems for completely positive maps

We prove lifting theorems for completely positive maps going out of exact C^* -algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if \mathsf X is a second countable topological space, \mathfrak A and \mathfrak B are separable, nuclear C^* -algebras over \mathsf X , and the action of \mathsf X on \mathfrak A is continuous, then E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B) naturally. As an application, we show that a separable, nuclear, strongly purely infinite C^* -algebra \mathfrak A absorbs a strongly self-absorbing C^* -algebra \mathscr D if and only if \mathfrak I and \mathfrak I\otimes \mathscr D are KK -equivalent for every two-sided, closed ideal \mathfrak I in \mathfrak A . In particular, if \mathfrak A is separable, nuclear, and strongly purely infinite, then \mathfrak A \otimes \mathcal O_2 \cong \mathfrak A if and only if every two-sided, closed ideal in \mathfrak A is KK -equivalent to zero.

Valdivia's lifting theorem for non-metrizable spaces

Valdivia's lifting theorem of (pre) compact sets and convergent (respectively, Cauchy) sequences from a quasi-(LB) space to a metrizable, strictly barrelled space is extended to a strictly larger collection of range spaces. Specifically, we assume that the range space has a sequential web structure and do not require that it be strictly barrelled, and need not even be barrelled. Distinguishing examples are provided that include natural constructions of range spaces connected with applications, such as DΓ′, the space of distributions having their wavefront sets in a specified closed cone Γ. We observe that the same and other examples could also serve as domain spaces for Valdivia's closed graph theorem, revealing a much wider collection of domain spaces that can be used in that result.

Stone dualities from opfibrations

Stone dualities are dual equivalences between certain categories of algebras and those of topological spaces. A Stone duality is often derived from a dual adjunction between such categories by cutting down unnecessary objects. This dual adjunction is called the fundamental adjunction of the duality, but building it often requires concrete topological arguments. The aim of this paper is to construct fundamental adjunctions generically using (op)fibered category theory. This paper defines an abstract notion of formal spaces (including ordinary topological spaces as the leading example), and gives a construction of a fundamental adjunction between the category of algebras and the category of corresponding formal spaces. Moreover, prove an Adjoint Lifting Theorem in the setting of near 2-fibrations, and construct the fundamental adjunction of Priestley duality.