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In mathematics, the Langlands–Deligne local constant (or local Artin root number up to an elementary 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(ρ''v'', ''s'', ψ''v'') of local constants ε(ρ''v'', ''s'', ψ''v'') associated to primes ''v''. Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis. proved the existence of the local constant ε(ρ''v'', ''s'', ψ''v'') up to sign. The original proof of the existence of the local constants by used local methods and was rather long and complicated, and never published. later discovered a simpler proof using global methods. ==Properties== The local constants ε(ρ, ''s'', ψ''E'') depend on a representation ρ of the Weil group and a choice of character ψ''E'' of the additive group of ''E''. They satisfy the following conditions: *If ρ is 1-dimensional then ε(ρ, ''s'', ψ''E'') 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'', ψ''E'') = ε(ρ1, ''s'', ψ''E'')ε(ρ2, ''s'', ψ''E''). As a result, ε(ρ, ''s'', ψ''E'') 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'', ψ''E'') = ε(Ind''E''/''K''ρ, ''s'', ψ''K'') Brauer's theorem on induced characters implies that these three properties characterize the local constants. showed that the local constants are trivial for real (orthogonal) representations of the Weil group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Langlands–Deligne local constant」の詳細全文を読む スポンサード リンク
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