L-function

Examples

This section provides an overview of basic examples of L-functions.

Riemann Zeta Function

The simplest example of an L-function, and at the same time the fundamental basis for any definition of the term L-function based on the work of Leonhard Euler, is the Riemann zeta function ${\displaystyle \zeta }$.

One of the possible “arithmetic objects” ${\displaystyle f}$ in the sense of the definition approach by Iwaniec and Kowalski, to which this L-function can be assigned, is the field ${\displaystyle \mathbb{Q}}$ of the rational numbers. Its Dirichlet series

${\displaystyle \sum _{n\in \mathbb{N}}\frac{1}{n^s},}$

that is

${\displaystyle \lambda (\mathbb{Q},n)=1}$

for all ${\displaystyle n\in \mathbb{N}}$, converges absolutely for ${\displaystyle \mathrm{\Re }(s)>1}$. Together with its Euler product, which is also absolutely convergent, the following holds for ${\displaystyle \mathrm{\Re }(s)>1}$:

${\displaystyle \zeta (s)=L(\mathbb{Q},s)=\sum _{n\in \mathbb{N}}\frac{1}{n^s}=\prod _{p\in \mathbb{P}}(1-p^{-s}{)}^{-1}.}$

Since all ${\displaystyle \lambda (\mathbb{Q},n)}$ are real, namely equal to 1, ${\displaystyle \zeta (s)}$ is self-dual. The object dual to ${\displaystyle f=\mathbb{Q}}$ is therefore also ${\displaystyle \mathbb{Q}}$, so ${\displaystyle \overline{f}=\mathbb{Q}}$.

The degree of the Euler product of the Riemann zeta function is

${\displaystyle d=1}$.

For its local parameters at ${\displaystyle p}$, the following holds:

${\displaystyle \alpha (\mathbb{Q},p)=1}$

for all ${\displaystyle p\in \mathbb{P}}$. Usually, the following gamma factor is used for the Riemann zeta function:

${\displaystyle \gamma (\mathbb{Q},s)={\pi }^{-\frac{s}{2}}\mathrm{\Gamma }\left(\frac{s}{2}\right).}$

The local parameter ${\displaystyle \kappa }$ at infinity is therefore 0. The conductor of ${\displaystyle \zeta }$ is

${\displaystyle q(\mathbb{Q})=1}$,

so that the complete Riemann zeta function takes the form

${\displaystyle \mathrm{\Lambda }(\mathbb{Q},s):=\gamma (\mathbb{Q},s)L(\mathbb{Q},s)={\pi }^{-\frac{s}{2}}\mathrm{\Gamma }\left(\frac{s}{2}\right)\zeta (s)}$

This definition is only valid for ${\displaystyle \mathrm{\Re }(s)>1}$, since the Riemann zeta function can be defined via its Dirichlet series or its Euler product only in this half-plane. However, the complete Riemann zeta function has an analytic continuation to a meromorphic function on the entire complex plane. This extension is holomorphic except for two simple poles at ${\displaystyle s=0}$ and ${\displaystyle s=1}$ with residues of −1 and 1, respectively. If we also denote the continued, complete Riemann zeta function by ${\displaystyle \mathrm{\Lambda }}$, then it satisfies the root number

${\displaystyle ϵ(\mathbb{Q})=1}$

the functional equation

${\displaystyle \mathrm{\Lambda }(\mathbb{Q},s)=\mathrm{\Lambda }(\mathbb{Q},1-s).}$

Thus, the Riemann zeta function, initially defined only for ${\displaystyle \mathrm{\Re }(s)>1}$ via its Dirichlet series or Euler product, now has an analytic continuation to a meromorphic function on ${\displaystyle \mathbb{C}}$, which is undefined only at ${\displaystyle s=1}$, since it has a simple pole with the residue 1 there. If we retain the notation ${\displaystyle \zeta }$ for the continued Riemann zeta function as well, it satisfies the functional equation

${\displaystyle {\pi }^{-\frac{s}{2}}\mathrm{\Gamma }\left(\frac{s}{2}\right)\zeta (s)={\pi }^{-\frac{1-s}{2}}\mathrm{\Gamma }\left(\frac{1-s}{2}\right)\zeta (1-s).}$

The (analytically extended) Riemann zeta function raises one of the most important questions in analytic number theory: the question of the exact location of its so-called non-trivial zeros. These lie in the “critical strip” ${\displaystyle 0<\mathrm{\Re }(s)<1}$. The Riemann Hypothesis from 1859—which remains neither proven nor disproven to this day—posits that all non-trivial zeros of the Riemann zeta function have a real part of ${\displaystyle 1/2}$. A proof of this hypothesis would allow for particularly accurate estimates of the distribution of prime numbers.

Dirichlet L-functions

The closest relatives of the Riemann zeta function are the Dirichlet L-functions, which include the Riemann zeta function as a special case. While in the Dirichlet series associated with the Riemann zeta function all ${\displaystyle \lambda }$ coefficients are equal to 1, in Dirichlet L-functions these are defined using a Dirichlet character. They thus take complex values with absolute value 1 or are equal to 0. Let ${\displaystyle m\in \mathbb{N}}$ be an integer, and let ${\displaystyle \chi }$ be a Dirichlet character modulo ${\displaystyle m}$:

${\displaystyle \chi :(\mathbb{Z}/m{)}^{\times }⟶S^1:=\{z\in \mathbb{C}:|z|=1\}}$

is given, i.e., a group homomorphism from the group of elements that are invertible with respect to multiplication in the residue class ring ${\displaystyle \mathbb{Z}/m}$ to the circle group ${\displaystyle S^1}$ of the complex numbers with absolute value 1. Such a Dirichlet character ${\displaystyle \chi }$ is called primitive, and ${\displaystyle m}$ is the generator of ${\displaystyle \chi }$ if it is not already given by a composition

${\displaystyle (\mathbb{Z}/m{)}^{\times }⟶(\mathbb{Z}/m^{\prime }{)}^{\times }\stackrel{{\chi }^{\prime }}{⟶}{S}^1}$

follows from a Dirichlet character ${\displaystyle {\chi }^{\prime }}$ modulo ${\displaystyle m^{\prime }}$ with a proper divisor ${\displaystyle m^{\prime }}$ of ${\displaystyle m}$. Using such a Dirichlet character ${\displaystyle \chi }$, one defines the following mapping, which is also denoted by ${\displaystyle \chi }$ and referred to as a Dirichlet character modulo ${\displaystyle m}$:

${\displaystyle \chi :\mathbb{Z}\to \mathbb{C},\chi (n)=\{\begin{array}{ll}\chi (n\mathrm{mod}m)& \text{if}\gcd (n,m)=1\\ 0& \text{if}\gcd (n,m)>1.\end{array}}$

The trivial Dirichlet characters ${\displaystyle {\chi }^0}$ modulo ${\displaystyle m}$ take the value 1 if ${\displaystyle \gcd (n,m)=1}$, and 0 otherwise. The trivial Dirichlet character modulo 1 is called the principal character. It satisfies ${\displaystyle \chi (n)=1}$ for all ${\displaystyle n\in \mathbb{N}}$.

Now, if ${\displaystyle \chi :\mathbb{Z}\to \mathbb{C}}$ is a primitive Dirichlet character modulo ${\displaystyle m}$, then an L-function is assigned to this arithmetic object ${\displaystyle \chi }$ as follows: With

${\displaystyle \lambda (\chi ,n):=\chi (n)}$

the Dirichlet series (also called the Dirichlet L-series)

${\displaystyle L(\chi ,s):=\sum _{n\in \mathbb{N}}\frac{\lambda (\chi ,n)}{n^s}=\sum _{n\in \mathbb{N}}\frac{\chi (n)}{n^s}}$

for ${\displaystyle \mathrm{\Re }(s)>1}$ is absolute. With the local parameters for ${\displaystyle p}$

${\displaystyle \alpha (\chi ,p):=\chi (p)}$

this also holds for the corresponding Euler product, and we have the identity

${\displaystyle L(\chi ,s)=\sum _{n\in \mathbb{N}}\frac{\chi (n)}{n^s}=\prod _{p\in \mathbb{P}}(1-\chi (p)p^{-s}{)}^{-1}}$

for ${\displaystyle \mathrm{\Re }(s)>1}$. As with the Riemann zeta function,

${\displaystyle d=1}$

is the order of the Euler product. If we set ${\displaystyle \kappa =0}$ when ${\displaystyle \chi (-1)=1}$ (in this case, ${\displaystyle \chi }$ is called “even”), and ${\displaystyle \kappa =1}$ if ${\displaystyle \chi (-1)=-1}$ (in this case, ${\displaystyle \chi }$ is called “odd”), then

${\displaystyle \gamma (\chi ,s)={\pi }^{-\frac{s}{2}}\mathrm{\Gamma }\left(\frac{s+\kappa }{2}\right)}$

the gamma factor associated with ${\displaystyle \chi }$. That ${\displaystyle \kappa \in \{0,1\}}$ is thus the local parameter at the infinite prime point. The leader ${\displaystyle m}$ of the primitive Dirichlet character ${\displaystyle \chi }$ is also the leader of the Dirichlet L-function:

${\displaystyle q(\chi )=m}$.

The complete Dirichlet L-function thus has the form

${\displaystyle \mathrm{\Lambda }(\chi ,s):=q(\chi {)}^{\frac{s}{2}}\gamma (\chi ,s)L(\chi ,s)={\left(\frac{m}{\pi }\right)}^{\frac{s}{2}}\mathrm{\Gamma }\left(\frac{s+\kappa }{2}\right)\sum _{n\in \mathbb{N}}\frac{\chi (n)}{n^s},}$

A definition that holds only for ${\displaystyle \mathrm{\Re }(s)>1}$, since the Dirichlet series used converges only there. However, such a complete Dirichlet L-function can be analytically extended to ${\displaystyle \mathbb{C}}$. This results in an integral function if ${\displaystyle \chi }$ is a non-trivial Dirichlet character. Otherwise, the extended function has a simple pole at ${\displaystyle s=1}$ with residue 1. The object dual to ${\displaystyle \chi }$ is ${\displaystyle \overline{\chi }}$, i.e., the Dirichlet character obtained from ${\displaystyle \chi }$ by complex conjugation of the function values of ${\displaystyle \chi }$, i.e.,

${\displaystyle \overline{\chi }(n)=\overline{\chi (n)}}$

for all ${\displaystyle n\in \mathbb{N}}$. The root number ${\displaystyle ϵ(\chi )}$ can be determined using the Gaussian sum

${\displaystyle \tau (\chi )=\sum _{x\mathrm{mod}q(\chi )}\chi (x)\exp (2\pi ix/q(\chi ))}$

where the summation extends over all residue classes modulo the generator ${\displaystyle q(\chi )=m}$ and ${\displaystyle \pi }$ denotes the circular number, ${\displaystyle i}$ denotes the imaginary unit, and ${\displaystyle \exp }$ denotes the exponential function. With

${\displaystyle ϵ(\chi )=\{\begin{array}{ll}\frac{\tau (\chi )}{\sqrt{q(\chi )}}& \text{if}\chi (-1)=1\\ \frac{\tau (\chi )}{i\sqrt{q(\chi )}}& \text{if}\chi (-1)=-1\end{array}}$

then the extended, complete Dirichlet L-function satisfies the functional equation

${\displaystyle \mathrm{\Lambda }(\chi ,s)=ϵ(\chi )\mathrm{\Lambda }(\overline{\chi },1-s).}$

As required by the root number theorem, ${\displaystyle |ϵ(\chi )|=1}$, since ${\displaystyle |\tau (\chi )|=\sqrt{q(\chi )}}$. The Dirichlet L-functions include the Riemann zeta function, since it arises from the trivial Dirichlet character modulo 1, i.e., the principal character.

In 1837, the German mathematician Peter Gustav Dirichlet used the Dirichlet L-functions, named after him, to prove the Dirichlet's prime number theorem, according to which in every arithmetic sequence (also called an arithmetic progression)

${\displaystyle a,a\pm n,a\pm 2n,a\pm 3n,\dots ,\text{ with }\gcd (a,n)=1,}$

where ${\displaystyle a,n\in \mathbb{N}}$

Namely, that in every residue class ${\displaystyle a\mathrm{mod}n}$, there are infinitely many prime numbers. The decisive argument in the proof of Dirichlet’s prime number theorem is the realization that ${\displaystyle \mathrm{\Lambda }(\chi ,1)\ne 0}$ holds for every non-trivial Dirichlet character ${\displaystyle \chi }$.

Dedekind L-functions

The Riemann zeta function is defined on the field ${\displaystyle \mathbb{Q}}$ of rational numbers, the simplest algebraic number field. Dedekind L-functions generalize this reference to arbitrary algebraic number fields, i.e., finite field extensions of ${\displaystyle \mathbb{Q}}$, such as ${\displaystyle \mathbb{Q}(\sqrt[3]{2})}$. Let ${\displaystyle K}$ be an algebraic number field and let ${\displaystyle n_K=[K:\mathbb{Q}]\in \mathbb{N}}$ be its degree of extension over ${\displaystyle \mathbb{Q}}$. Let ${\displaystyle {\mathcal{O}}_K}$ be its integral domain and ${\displaystyle d_K\in \mathbb{Z}}$ its discriminant. Furthermore, let ${\displaystyle n_1\in {\mathbb{N}}_0}$ be the number of real embeddings and ${\displaystyle n_2\in {\mathbb{N}}_0}$ the number of pairs of complex embeddings of ${\displaystyle K}$. Thus, ${\displaystyle n_K=n_1+2n_2}$.

The Dedekind L-function (also known as the Dedekind zeta function) with respect to ${\displaystyle K}$ is defined for ${\displaystyle \mathrm{\Re }(s)>1}$ by

${\displaystyle {\zeta }_K(s):=L(K,s):=\sum _{0\ne \mathfrak{a}\subset {\mathcal{O}}_K}\frac{1}{\mathcal{N}(\mathfrak{a}{)}^s}.}$

In the sum, ${\displaystyle \mathfrak{a}}$ runs through all integer ideals of ${\displaystyle {\mathcal{O}}_K}$ that are distinct from the zero ideal ${\displaystyle \{0\}}$. ${\displaystyle \mathcal{N}(\mathfrak{a})\in \mathbb{N}}$ denotes the absolute norm of ${\displaystyle \mathfrak{a}}$. The coefficients of the Dirichlet series

${\displaystyle \sum _{0\ne \mathfrak{a}\subset {\mathcal{O}}_K}\frac{1}{\mathcal{N}(\mathfrak{a}{)}^s}=\sum _{n\in \mathbb{N}}\frac{\lambda (K,n)}{n^s}}$

are thus

${\displaystyle \lambda (K,n)=\mathrm{\#}\{0\ne \mathfrak{a}\subset {\mathcal{O}}_K\mid \mathcal{N}(\mathfrak{a})=n\}\in {\mathbb{N}}_0.}$

For every ${\displaystyle n\in \mathbb{N}}$, they specify the number of integral ideals of ${\displaystyle {\mathcal{O}}_K}$ with absolute norm ${\displaystyle n}$. In particular, all coefficients ${\displaystyle \lambda (K,n)}$ are real, and therefore ${\displaystyle L(K,s)}$ is self-dual. That Dirichlet series converges absolutely for ${\displaystyle \mathrm{\Re }(s)>1}$, as does the corresponding Euler product

${\displaystyle \prod _{0\ne \mathfrak{p}\subset {\mathcal{O}}_K}\frac{1}{1-\mathcal{N}(\mathfrak{p}{)}^{-s}}.}$

Here, the product extends over all prime ideals ${\displaystyle \mathfrak{p}}$ of ${\displaystyle {\mathcal{O}}_K}$ that are distinct from the zero ideal. For ${\displaystyle \mathrm{\Re }(s)>1}$, the following identity holds

${\displaystyle L(K,s)=\sum _{0\ne \mathfrak{a}\subset {\mathcal{O}}_K}\frac{1}{\mathcal{N}(\mathfrak{a}{)}^s}=\sum _{n\in \mathbb{N}}\frac{\lambda (K,n)}{n^s}=\prod _{0\ne \mathfrak{p}\subset {\mathcal{O}}_K}\frac{1}{1-\mathcal{N}(\mathfrak{p}{)}^{-s}}.}$

This form of the Euler product does not yet show the individual Euler factors ${\displaystyle (1-{\alpha }_1(K,p)p^{-s}{)}^{-1}\cdot \dots \cdot (1-{\alpha }_d(K,p)p^{-s}{)}^{-1}}$. In any case, the degree of the Euler product is equal to the degree of the field extension ${\displaystyle K/\mathbb{Q}}$:

${\displaystyle d=[K:\mathbb{Q}]=n_K=n_1+2n_2.}$

The local parameters ${\displaystyle {\alpha }_j(K,p)}$ depend on the decomposition behavior of the ideals ${\displaystyle (p):=p{\mathcal{O}}_K}$

every ideal ${\displaystyle (p)}$ has a “prime ideal decomposition” that is unique up to the order of the factors

${\displaystyle (p)=\prod _{\mathfrak{p}}{\mathfrak{p}}^{e_{\mathfrak{p}}}}$

in prime ideals ${\displaystyle 0\ne \mathfrak{p}\subset {\mathcal{O}}_K}$, where the following holds: ${\displaystyle e_{\mathfrak{p}}\in {\mathbb{N}}_0}$ and ${\displaystyle e_{\mathfrak{p}}>0}$ for only finitely many prime ideals ${\displaystyle \mathfrak{p}}$. For at most ${\displaystyle n_K}$ prime ideals ${\displaystyle \mathfrak{p}}$, ${\displaystyle e_{\mathfrak{p}}>0}$ may hold. Such ${\displaystyle \mathfrak{p}}$ divide ${\displaystyle (p)}$, and we write ${\displaystyle \mathfrak{p}|(p)}$. The exponent ${\displaystyle e_{\mathfrak{p}}}$ in the prime ideal decomposition of ${\displaystyle (p)}$ is called the branching index of ${\displaystyle \mathfrak{p}}$ over ${\displaystyle p}$. If ${\displaystyle \mathfrak{p}|(p)}$, then ${\displaystyle \mathcal{N}(\mathfrak{p})=p^{f_{\mathfrak{p}}}}$ for some ${\displaystyle f_{\mathfrak{p}}\in \mathbb{N}}$, which is called the inertia index of ${\displaystyle \mathfrak{p}}$ over ${\displaystyle p}$. For every ${\displaystyle p\in \mathbb{P}}$, the branching and inertia indices associated with the ideal ${\displaystyle (p)}$ satisfy the following relation with respect to the degree of ${\displaystyle K/\mathbb{Q}}$:

${\displaystyle \sum _{\mathfrak{p}|(p)}{e}_{\mathfrak{p}}{f}_{\mathfrak{p}}=n_K.}$

Using the knowledge of the inertia indices for every ${\displaystyle p\in \mathbb{P}}$, the local parameters ${\displaystyle {\alpha }_j(K,p)}$ can now be determined, namely via the factors ${\displaystyle (1-p^{-s{f}_{\mathfrak{p}}}{)}^{-1}}$ in the identity

${\displaystyle \prod _{0\ne \mathfrak{p}\subset {\mathcal{O}}_K}\frac{1}{1-\mathcal{N}(\mathfrak{p}{)}^{-s}}=\prod _{p\in \mathbb{P}}\prod _{\mathfrak{p}|(p)}(1-p^{-s{f}_{\mathfrak{p}}}{)}^{-1},}$

by factoring the polynomials ${\displaystyle X^{f_{\mathfrak{p}}}-1}$ in the polynomial ring ${\displaystyle \mathbb{C}[X]}$.

The gamma factor with respect to ${\displaystyle L(K,s)}$ is

${\displaystyle \gamma (K,s)={\pi }^{-\frac{s\cdot n_K}{2}}\mathrm{\Gamma }{\left(\frac{s}{2}\right)}^{n_1+n_2}\mathrm{\Gamma }{\left(\frac{s+1}{2}\right)}^{n_2}.}$

The value of the discriminant of ${\displaystyle K}$ is the conductor of ${\displaystyle L(K,s)}$:

${\displaystyle q(K)=|d_K|.}$

Thus, the complete L-function of ${\displaystyle K}$ for ${\displaystyle \mathrm{\Re }(s)>1}$ is given by

${\displaystyle \mathrm{\Lambda }(K,s):=q(K{)}^{\frac{s}{2}}\gamma (K,s)L(K,s)={\left(\frac{|d_K|}{{\pi }^{n_K}}\right)}^{\frac{s}{2}}\mathrm{\Gamma }{\left(\frac{s}{2}\right)}^{n_1+n_2}\mathrm{\Gamma }{\left(\frac{s+1}{2}\right)}^{n_2}\sum _{0\ne \mathfrak{a}\subset {\mathcal{O}}_K}\frac{1}{\mathcal{N}(\mathfrak{a}{)}^s}.}$

This function has an analytic continuation to the complex plane with simple poles at ${\displaystyle s=0}$ and ${\displaystyle s=1}$, and residues there of ${\displaystyle -\frac{2^{r}hR}{w}}$ and ${\displaystyle \frac{2^{r}hR}{w}}$, respectively. Here, ${\displaystyle r=n_1+n_2}$ is the number of infinite digits, ${\displaystyle h\in \mathbb{N}}$ is the class number, ${\displaystyle R\in \mathbb{R}}$ is the regulator of ${\displaystyle K}$, and ${\displaystyle w\in \mathbb{N}}$ is the number of unit roots in ${\displaystyle K}$.

Dedekind L-functions always have a root of 1:

${\displaystyle ϵ(K)=1.}$

Thus, the analytically extended, complete L-function of ${\displaystyle K}$ satisfies the functional equation

${\displaystyle \mathrm{\Lambda }(K,s)=\mathrm{\Lambda }(K,1-s).}$

The analytically extended function ${\displaystyle \mathrm{\Lambda }(K,s)}$ now also allows for the analytic extension of ${\displaystyle L(K,s)}$, namely via the definition

${\displaystyle L(K,s):=\frac{\mathrm{\Lambda }(K,s)}{|d_k{|}^{\frac{s}{2}}\gamma (K,s)}={\left(\frac{{\pi }^{n_K}}{|d_K|}\right)}^{\frac{s}{2}}\cdot \frac{\mathrm{\Lambda }(K,s)}{\mathrm{\Gamma }{\left(\frac{s}{2}\right)}^{n_1+n_2}\mathrm{\Gamma }{\left(\frac{s+1}{2}\right)}^{n_2}}.}$

This makes ${\displaystyle L(K,s)}$ a meromorphic function on ${\displaystyle \mathbb{C}}$ with a simple pole at ${\displaystyle s=1}$. One of its fascinating properties is the so-called analytic class number formula, according to which the residue of ${\displaystyle L(K,s)}$ at ${\displaystyle s=1}$ takes the following form:

${\displaystyle {\mathrm{Res}}_{s=1}L(K,s)=\frac{2^{n_1}(2\pi {)}^{n_2}}{w\sqrt{|d_K|}}hR.}$

Conjectural information

One can list characteristics of known examples of L-functions that one would wish to see generalized:

• location of zeros and poles;
• functional equation, with respect to some vertical line Re(s) = constant;
• interesting values at integers related to quantities from algebraic K-theory.

Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta function connects through its values at positive even integers (and negative odd integers) to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules.

The statistics of the zero distributions are of interest because of their connection to problems like the generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest. The fractal structure of the distributions has been studied using rescaled range analysis. The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function, and also for the zeros of other L-functions of different orders and conductors.

Birch and Swinnerton-Dyer conjecture

One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s. It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another global field): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of L-functions. This was something like a paradigm example of the nascent theory of L-functions.

Rise of the general theory

This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin L-functions, which, like Hecke L-functions, were defined several decades earlier, and to L-functions attached to general automorphic representations.

Gradually it became clearer in what sense the construction of Hasse–Weil zeta functions might be made to work to provide valid L-functions, in the analytic sense: there should be some input from analysis, which meant automorphic analysis. The general case now unifies at a conceptual level a number of different research programs.