Unitary Completion Research Articles

A Note on Integral Structures in Some Locally Algebraic Representations of GL2

In the p-adic local Langlands correspondence for GL2 (ℚp), the following theorem of Berger and Breuil has played an important role: the locally algebraic representations of GL2(ℚp) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither (ϕ, Γ)-module techniques nor global methods. When F is a finite extension of ℚp, we also get a simpler proof of a theorem of Vigneras for the existence of integral structures for (locally algebraic) special series and for (smooth) tamely ramified principal series.

Unitarity bounds on charged/neutral state mass ratios

In this letter, we study the implications of unitary completion of quantum gravity on the low energy spectrums, through an infinite set of unitarity bounds on the forward-limit scattering amplitudes. In three dimensions, we find that light states with charge-to-mass ratio $z$ greater than $1$ can only be consistent if there exists other light states, preferably neutral. Applied to the compactification of the Standard Model, where the low energy couplings are dominated by the electron with $|z|\sim 10^{22}$, this provides a novel understanding of the need for light neutrinos.

Unitary completions of locally algebraic and locally analytic principal series of p-adic GL2

We generalize a result of Emerton on the relationship between unitary completions of locally [Formula: see text]-analytic and locally [Formula: see text]-algebraic principal series representations induced from certain locally [Formula: see text]-algebraic characters of the diagonal torus of [Formula: see text], where [Formula: see text] is a finite extension of [Formula: see text]. Namely, under a non-critical slope hypothesis on the character being induced, the map on universal unitary completions arising from the inclusion of the locally algebraic induction into the locally analytic induction is a topological isomorphism. (Emerton proved this result for [Formula: see text].) The main ingredients in carrying out a “several-variable” version of Emerton’s argument are the description of the local convex space of locally [Formula: see text]-analytic functions on the group [Formula: see text] in terms of the embeddings of [Formula: see text] into our [Formula: see text]-adic coefficient field, and a generalization by Breuil of the classical result of Amice-Vélu and Vishik on “tempered distributions” on [Formula: see text].

On unitary extensions and unitary completions of topological monoids

Abstract The concept of a unitary Cauchy net in an arbitrary Hausdorff topological monoid generalizes the concept of a fundamental sequence of reals. We construct extensions of this monoid where all its unitary Cauchy nets converge.

Extensions de représentations de de Rham et vecteurs localement algébriques

Let${\rm\Pi}$be an irreducible unitary completion of a locally algebraic$\text{GL}_{2}(\mathbf{Q}_{p})$-representation. We describe those first-order deformations of${\rm\Pi}$which are themselves completions of a locally algebraic representation. This answers a question of Paškūnas and has direct applications to the Breuil–Mézard conjecture.

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL2(Q p ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL2(Q p ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.

Admissible unitary completions of locally ℚ_{𝕡}-rational representations of 𝔾𝕃₂(𝔽)

Let F F be a finite extension of Q p \mathbb {Q}_p , p > 2 p>2 . We construct admissible unitary completions of certain representations of G L 2 ( F ) \mathrm {GL}_2(F) on L L -vector spaces, where L L is a finite extension of F F . When F = Q p F=\mathbb {Q}_p using the results of Berger, Breuil and Colmez we obtain some results about lifting 2 2 -dimensional mod p p representations of the absolute Galois group of Q p \mathbb {Q}_p to crystabelline representations with given Hodge-Tate weights.

The unitary completion and QR iterations for a class of structured matrices

We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower triangular part of a special form. For such a specified part the unitary completion is a structured matrix, and we derive in this paper the formulas for its structure. Next we apply the unitary completion method to the solution of the eigenvalue problem for a class of structured matrices via structured QR iterations.

P -adic L -functions and unitary completions of representations of P -adic reductive groups

In the first half of the article, we introduce the notion of the universal unitary completion of a continuous representation of a p -adic reductive group on a locally convex p -adic vector space, and we prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil [4] in connection with his work on a possible p -adic local Langlands correspondence for GL 2 , and we relate our construction to certain conjectures of Breuil in [4, Sec. 1.3] for the group GL 2 ( ℚ p ) . In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a noncritical slope condition (see Prop. 2.5). In the second half of the article, we consider a certain unitary Banach space representation of GL 2 ( ℚ p ) obtained by p -adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic, parabolically induced representations of GL 2 ( ℚ p ) which arise from classical finite-slope newforms have a nontrivial universal unitary completion (verifying a conjecture of Breuil in [4, Sec. 1.3] for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of p -adic L -functions attached to p -stabilized newforms of noncritical slope. Combining our construction with the representation-theoretic definition of L -invariants implicit in [5, Cor. 1.1.7], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture in [18, p. 46] (in terms of Breuil's definition of the L -invariant)

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.

On the unitary completion of a matrix

On the unitary completion of a matrix