nLab L-function




LL-functions are certain meromorphic functions generalizing the Riemann zeta function. They are typically defined on parts of the complex plane by a power series expressions – called the LL-series – which converges in that region, and then meromorphically extended to all of the complex plane by analytic continuation.

The most canonically defined class of examples of L-functions are the Artin L-functions defined for any Galois representation σ:GalGL n()\sigma \colon Gal \longrightarrow GL_n(\mathbb{C}) as the Euler products of, essentially, the characteristic polynomials of all the Frobenius homomorphisms acting via σ\sigma.

Another common example of an L-function comes from modular forms. If we have a cusp form ff of weight kk, with Fourier expansion

f(z)= n=1 a ne 2πinzf(z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i n z}

the associated L-function L f(s)L_{f}(s) is given by

L f(s)= n a nn s.L_{f}(s)=\sum_{n}^{\infty}a_{n}n^{-s}.

The L-function L f(s)L_{f}(s) may also be obtained as the Mellin transform of ff. This is an example of an automorphic L-function, which more generally is not as easy to construct.

Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:

  1. for 1-dimensional Galois representations σ\sigma (hence for n=1n = 1) Artin reciprocity produces for each σ\sigma a Dirichlet character, or more generally a Hecke character? χ\chi, and therefrom is built a Dirichlet L-function or Hecke L-function L χL_\chi, respectively, which equals the corresponding Artin L-function L σL_\sigma;

  2. for general nn-dimensional Galois representations σ\sigma the conjecture of Langlands correspondence states that there is an automorphic representation π\pi corresponding to σ\sigma and an automorphic L-function L πL_\pi built from that, which equalso the Artin L-function L σL_\sigma.

L-functions typically satisfy analogs of all the special properties enjoyed by the Riemann zeta function, such as satisfying a “functional equation” which asserts invariance under modular transformations of the parameter.

The generalized Riemann conjecture is concerned with zeros of the Dedekind zeta function for which the L-series (the Dirichlet L-function) is complicated from the classical Riemann case by the presence of the additional parameter, the Dirichlet character.

context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of θ(z,)\theta(\mathbf{z},-))eta function η\etaspecial values of L-functions
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued trace of Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy 12L z (0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\mathbf{z}, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number \cdot regulator
arithmetic geometry for \mathbb{Q}Jacobi theta function (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=χ\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\mathbf{z} , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function



Some history is in

  • James W. Cogdell, L-functions and non-abelian class eld theory, from Artin to Langlands, 2012 (pdf)

Last revised on July 2, 2022 at 19:10:30. See the history of this page for a list of all contributions to it.