Langlands–Deligne local constant
{{short description|Elementary function in mathematics}}
In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor{{cite journal |title=Elliptic curves and local ϵ-factors |last1=Kramer |first1=K. |last2=Tunnell |first2= J. |author-link2= Jerrold B. Tunnell |journal=Compositio Mathematica |volume=46 |year=1982 |issue=3 |pages=307–352}} or local Artin root number (up to an elementary real function of s), is an elementary function associated with a representation of the Weil group of a local field. The functional equation
:L(ρ,s) = ε(ρ,s)L(ρ∨,1−s)
of an Artin L-function has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product
:ε(ρ,s) = Π ε(ρv, s, ψv)
of local constants ε(ρv, s, ψv) associated to primes v.
Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis.
{{harvtxt|Dwork|1956}} proved the existence of the local constant ε(ρv, s, ψv) up to sign.
The original proof of the existence of the local constants by {{harvtxt|Langlands|1970}} used local methods and was rather long and complicated, and never published. {{harvtxt|Deligne|1973}} later discovered a simpler proof using global methods.
Properties
The local constants ε(ρ, s, ψE) depend on a representation ρ of the Weil group and a choice of character ψE of the additive group of E. They satisfy the following conditions:
- If ρ is 1-dimensional then ε(ρ, s, ψE) is the constant associated to it by Tate's thesis as the constant in the functional equation of the local L-function.
- ε(ρ1⊕ρ2, s, ψE) = ε(ρ1, s, ψE)ε(ρ2, s, ψE). As a result, ε(ρ, s, ψE) can also be defined for virtual representations ρ.
- If ρ is a virtual representation of dimension 0 and E contains K then ε(ρ, s, ψE) = ε(IndE/Kρ, s, ψK)
Brauer's theorem on induced characters implies that these three properties characterize the local constants.
{{harvtxt|Deligne|1976}} showed that the local constants are trivial for real (orthogonal) representations of the Weil group.
Notational conventions
There are several different conventions for denoting the local constants.
- The parameter s is redundant and can be combined with the representation ρ, because ε(ρ, s, ψE) = ε(ρ⊗||s, 0, ψE) for a suitable character ||.
- Deligne includes an extra parameter dx consisting of a choice of Haar measure on the local field. Other conventions omit this parameter by fixing a choice of Haar measure: either the Haar measure that is self dual with respect to ψ (used by Langlands), or the Haar measure that gives the integers of E measure 1. These different conventions differ by elementary terms that are positive real numbers.
References
{{Reflist}}
- {{Citation | last1=Bushnell | first1=Colin J. |authorlink1=Colin J. Bushnell |last2=Henniart | first2=Guy |authorlink2=Guy Henniart | title=The local Langlands conjecture for GL(2) | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-31486-8 | doi=10.1007/3-540-31511-X | mr=2234120 | year=2006 | volume=335}}
- {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/978-3-540-37855-6_7 | mr=0349635 | year=1973 | volume=349 | chapter=Les constantes des équations fonctionnelles des fonctions L | pages=501–597| isbn=978-3-540-06558-6 }}
- {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Les constantes locales de l'équation fonctionnelle de la fonction L d'Artin d'une représentation orthogonale | doi=10.1007/BF01390143 | mr=0506172 | year=1976 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=35 | pages=299–316| s2cid=119880957 }}
- {{Citation | last1=Dwork | first1=Bernard | title=On the Artin root number | jstor=2372524 | mr=0082476 | year=1956 | journal=American Journal of Mathematics | issn=0002-9327 | volume=78 | issue=2 | pages=444–472 | doi=10.2307/2372524}}
- {{citation|last=Langlands|first=Robert|series=Unpublished notes|url=http://publications.ias.edu/rpl/paper/61|title=On the functional equation of the Artin L-functions|year=1970}}
- {{Citation | last1=Tate | first1=John T. | editor1-last=Fröhlich | editor1-first=A. | title=Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) | url=https://books.google.com/books?id=_QDvAAAAMAAJ | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-268960-4 | mr=0457408 | year=1977 | chapter=Local constants | pages=89–131}}
- {{citation|last=Tate|first= J. |chapter=Number theoretic background |chapter-url=https://www.ams.org/online_bks/pspum332/ |title=Automorphic forms, representations, and L-functions Part 2 |pages= 3–26|series=Proc. Sympos. Pure Math.|volume= XXXIII|publisher= Amer. Math. Soc.|publication-place= Providence, R.I.|year=1979|isbn=0-8218-1435-4}}
External links
- {{Springer|first=R.|last= Perlis|id=a/a120270|title=Artin root numbers}}
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