First steps towards p-adic Langlands functoriality
Identifieur interne : 000A14 ( Main/Merge ); précédent : 000A13; suivant : 000A15First steps towards p-adic Langlands functoriality
Auteurs : Christophe Breuil [France] ; Peter Schneider [Allemagne]Source :
- Journal für die reine und angewandte Mathematik (Crelles Journal) [ 0075-4102 ] ; 2007-09-26.
Descripteurs français
- Wicri :
- topic : Recevabilité.
English descriptors
- KwdEn :
- Admissibility, Algebra, Algebraic, Algebraic representation, Analytic functions, Banach, Banach algebra, Base extension, Bcris, Borel subgroup, Breuil, Category ficl, Central character, Conjecture, Crystalline galois representations, Crystalline representations, Cyclotomic character, Dcris, Duke math, Embedding, Faithful tensor functor, Ficl, Fontaine, Frobenius, Functor, Functor dcris, Functoriality, Galois, Galois group, Galois representations, Galois side, Highest weight, Homomorphism, Indu, Inequality, Invariant norm, Irreducible, Isomorphic, Isomorphism, Isomorphism class, Isomorphism classes, Langlands, Langlands correspondence, Langlands functoriality, Local class, Ltration, Modl, Module, Nite, Nite extension, Nite rank, Other hand, Parabolic, Quotient, Rational representation, Reductive, Reductive group, Reductive groups, Repk, Representation, Resp, Rham representation, Satake, Satake isomorphism, Schneider, Semisimple, Semisimple part, Smooth part, Special case, Square root, Subgroup, Tannakian, Tannakian categories, Tensor, Unique model, Vall, Vall detk, Vector space, Weil, Weil representation.
- Teeft :
- Admissibility, Algebra, Algebraic, Algebraic representation, Analytic functions, Banach, Banach algebra, Base extension, Bcris, Borel subgroup, Breuil, Category ficl, Central character, Conjecture, Crystalline galois representations, Crystalline representations, Cyclotomic character, Dcris, Duke math, Embedding, Faithful tensor functor, Ficl, Fontaine, Frobenius, Functor, Functor dcris, Functoriality, Galois, Galois group, Galois representations, Galois side, Highest weight, Homomorphism, Indu, Inequality, Invariant norm, Irreducible, Isomorphic, Isomorphism, Isomorphism class, Isomorphism classes, Langlands, Langlands correspondence, Langlands functoriality, Local class, Ltration, Modl, Module, Nite, Nite extension, Nite rank, Other hand, Parabolic, Quotient, Rational representation, Reductive, Reductive group, Reductive groups, Repk, Representation, Resp, Rham representation, Satake, Satake isomorphism, Schneider, Semisimple, Semisimple part, Smooth part, Special case, Square root, Subgroup, Tannakian, Tannakian categories, Tensor, Unique model, Vall, Vall detk, Vector space, Weil, Weil representation.
Abstract
By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.
Url:
DOI: 10.1515/CRELLE.2007.070
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<term>Banach</term>
<term>Banach algebra</term>
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<term>Bcris</term>
<term>Borel subgroup</term>
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<term>Duke math</term>
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<term>Faithful tensor functor</term>
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<term>Isomorphism classes</term>
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<term>Langlands correspondence</term>
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<term>Ltration</term>
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<term>Parabolic</term>
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<term>Algebraic</term>
<term>Algebraic representation</term>
<term>Analytic functions</term>
<term>Banach</term>
<term>Banach algebra</term>
<term>Base extension</term>
<term>Bcris</term>
<term>Borel subgroup</term>
<term>Breuil</term>
<term>Category ficl</term>
<term>Central character</term>
<term>Conjecture</term>
<term>Crystalline galois representations</term>
<term>Crystalline representations</term>
<term>Cyclotomic character</term>
<term>Dcris</term>
<term>Duke math</term>
<term>Embedding</term>
<term>Faithful tensor functor</term>
<term>Ficl</term>
<term>Fontaine</term>
<term>Frobenius</term>
<term>Functor</term>
<term>Functor dcris</term>
<term>Functoriality</term>
<term>Galois</term>
<term>Galois group</term>
<term>Galois representations</term>
<term>Galois side</term>
<term>Highest weight</term>
<term>Homomorphism</term>
<term>Indu</term>
<term>Inequality</term>
<term>Invariant norm</term>
<term>Irreducible</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
<term>Isomorphism class</term>
<term>Isomorphism classes</term>
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<term>Langlands correspondence</term>
<term>Langlands functoriality</term>
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<term>Ltration</term>
<term>Modl</term>
<term>Module</term>
<term>Nite</term>
<term>Nite extension</term>
<term>Nite rank</term>
<term>Other hand</term>
<term>Parabolic</term>
<term>Quotient</term>
<term>Rational representation</term>
<term>Reductive</term>
<term>Reductive group</term>
<term>Reductive groups</term>
<term>Repk</term>
<term>Representation</term>
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<term>Rham representation</term>
<term>Satake</term>
<term>Satake isomorphism</term>
<term>Schneider</term>
<term>Semisimple</term>
<term>Semisimple part</term>
<term>Smooth part</term>
<term>Special case</term>
<term>Square root</term>
<term>Subgroup</term>
<term>Tannakian</term>
<term>Tannakian categories</term>
<term>Tensor</term>
<term>Unique model</term>
<term>Vall</term>
<term>Vall detk</term>
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<front><div type="abstract" xml:lang="en">By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.</div>
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