Contents

# Contents

## Idea

$L$-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 $L$-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 $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the Euler products of, essentially, the characteristic polynomials of all the Frobenius homomorphisms acting via $\sigma$.

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

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

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

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

The L-function $L_{f}(s)$ may also be obtained as the Mellin transform of $f$. 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 $\sigma$ (hence for $n = 1$) Artin reciprocity produces for each $\sigma$ a Dirichlet character, or more generally a Hecke character? $\chi$, and therefrom is built a Dirichlet L-function or Hecke L-function $L_\chi$, respectively, which equals the corresponding Artin L-function $L_\sigma$;

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

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 $\theta$zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)eta function $\eta$special values of L-functions
physics/2d CFTpartition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
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 $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$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 $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, 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 ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)Artin L-function of a Galois representation $\mathbf{z}$ , 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

## Examples

• Stephen Gelbart, section C starting on p. 14 (190) of An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)

• E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

• Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

Some history is in

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