Functional equation (L-function): Difference between revisions

Revision as of 17:39, 4 December 2009

Revision as of 06:02, 17 December 2009

Line 1:

{{Unreferenced|date=December 2009}}

In [[mathematics]], the [[L-function]]s of [[number theory]] are expected to have several characteristic properties, one of which is that they satisfy certain '''[[functional equation]]s'''. There is an elaborate theory of what these equations should be, much of which is still conjectural. For example, the [[Riemann zeta function]] has a functional equation relating its value at the [[complex number]] ''s'' with its value at 1 − ''s''. In every case this relates to some value ζζ(''s'') that is only defined by [[analytic continuation]] from the [[infinite series]] definition. That is, writing{{ndashNdash}}as is conventional{{ndashNdash}}σ for the real part of ''s'', the functional equation relates the cases

:σ > 1 and σ < 0,

Line 7 ⟶ 8:

:0 < σ < 1

in the ''critical strip'' to another such case, reflected in the line σ = &frac12;½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole [[complex plane]].

The functional equation in question for the Riemann zeta function takes the simple form

Line 19 ⟶ 20:

:<math>\Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^*)</math>

with χ a [[primitive Dirichlet character]], χ^* its complex conjugate, &Lambda;Λ the L-function multiplied by a gamma-factor, and &epsilon;ε a complex number of [[absolute value]] 1, of shape

:<math>G(\chi) \over {\left |G(\chi)\right \vert}</math>

where ''G''(χ) is a [[Gauss sum]] formed from χ. This equation has the same function on both sides if and only if χ is a ''real character'', taking values in {0,1,&minus;1}. Then &epsilon;ε must be 1 or &minus;1, and the case of the value &minus;1 would imply a zero of ''&Lambda;Λ''(''s'') at ''s'' = &frac12;½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such ''simple'' zero can exist (the function is ''even'' about the point).

A unified theory of such functional equations was given by [[Erich Hecke]], and the theory was taken up again in ''[[Tate's thesis]]'' by [[John Tate]]. Hecke found generalised characters of number fields, now called [[Hecke character]]s, for which his proof (based on [[theta function]]s) also worked. These characters and their associated L-functions are now understood to be strictly related to [[complex multiplication]], as the Dirichlet characters are to [[cyclotomic field]]s.

There are also functional equations for the [[local zeta-function]]s, arising at a fundamental level for the (analogue of) [[Poincaré duality]] in [[étale cohomology]]. The Euler products of the [[Hasse-Weil zeta-function]] for an [[algebraic variety]] ''V'' over a number field ''K'', formed by reducing ''modulo'' [[prime ideal]]s to get local zeta-functions, are conjectured to have a ''global'' functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from [[automorphic representation]] theory seems required to get the functional equation. The [[Taniyama-Shimura conjecture]] was a particular case of this as general theory. By relating the gamma-factor aspect to [[Hodge theory]], and detailed studies of the expected &epsilon;ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.

==See also==

*[[explicit formula (L-function)]]

*[[approximate functional equation]]

{{DEFAULTSORT:Functional Equation (L-Function)}}

[[Category:Zeta and L-functions]]

[[Category:Functional equations]]

[[Category:Articles lacking sources (Erik9bot)]]

[[fr:Équation fonctionnelle (fonction L)]]