Taxonomy of L-functions (awaiting review)

Taxonomy of L-functions

One can consider $L$-functions as autonomous objects, not necessarily arising from another more basic object. From that point of view it is natural to develop a taxonomy of $L$-functions, where only the properties of the $L$-function are used for the classification.

The most obvious invariant of an $L$-function is its degree. After the degree, it is not immediately obvious which properties of the $L$-function should be considered next, but we propose the following as a reasonable scheme:

Note: some of the names could be better. And the order is not canonical.

0) Degree.

The non-negative integer $d=J+2K$, where $J$ and $K$ refer respectively to the number of $\Gamma_\R$ and $\Gamma_\C$ factors in the functional equation. Recall that the $L$-function axioms determine $J$ and $K$ uniquely.

1) Signature.

The pair $(J,K)$.

1.5) $\Gamma$-data

The pair $(\{\mu_1,...,\mu_J\}, \{\nu_1,...,\nu_K\})$, where at this point we may wish to think of the $\mu_j$ and $\nu_k$ as depending on a parameter. For example, the two main classes of degree 2 $L$-functions have: $$ (\{i\lambda,-i\lambda\},\{\}) \ \ \ \ \ \ \ \ \ or \ \ \ \ \ \ \ \ \ (\{\},\{\frac{k-1}{2}\}), $$ where $\lambda$ is a positive real number and $k$ is an integer.

2) Is the $L$-function algebraic?

We call an $L$-function algebraic if its Dirichlet coefficients are algebraic numbers. This, conjecturally, is equivalent to each of:

a) $\mu_j$ and $\nu_k$ are real

b) The $L$-function is the $L$-function of a Galois representation.

c) The $L$-function is motivic

d) There exists a number field $K/\Q$, called the field of coefficients, and a integer $w$, called the motivic weight of the $L$-function, such that $a_n n^{w/2}$ is in integer of $K$ for all $n$. (Note that $w$ is chosen to be the minimal integer with this property.) (Note also that, conjecturally, $w=\min\{0,\max\{\nu_k\}\}$. In the arithmetic normalization, the functional equation relates $s$ to $w+1-s$.) (Note that for holomorphic $GL(2)$ cusp forms, motivic weight is one less than the "classical" weight. For Siegel modular forms, the "classical" weight has yet another relationship to the motivic weight.)

2') The motivic weight

2'') The field of coefficients

3) Level.

The non-negative integer $N$ appearing in the functional equation.

Note: Once the level and the specific numbers in the $\Gamma$-factors have been specified, then (conjecturally) there are only finitely many $L$-functions possible.

4) Central character.

The Euler product has the form $$ L(s) = \prod_p F_p(p^{-s}) $$ where $$ F_p(x) = 1+a_p x + \cdots + (-1)^d \chi(p) x^d. $$ The character $\chi$, which is a Dirichlet character mod~$N$, is called the central character of the $L$-function.

If $p$ is a good prime, then the roots of $F_p$ are called the Satake paramaters at $p$ for the $L$-funciton.

Note: if $L(s)=L(s,f)$ for some object $f$, then $f$ may have its own Satake parameters, which in general are different from the Satake parameters of the $L$-function. (This has to be the case, because $f$ probably has infinitely many $L$-functions associated to it, and each $L$-function has different Satake parameters.)

5) Sign.

The parameter $\varepsilon$ appearing in the functional equation. Also known as the “root number.”

The above classification exhausts the parameters in the functional equation and the Euler product, but there are further refinements based on the distribution of Dirichlet coefficients.

6) Self-dual?

Are the Dirichlet coefficients real?

7) Sato-Tate group

The Sato-Tate group of an $L$-function is a compact subgroup of a classical matrix group whose eigenvalues under Haar measure have the same distribution as the distribution of Satake parameters of the $L$-function. Further discussion below.

The Dirichlet coefficients $a_p$,...,$a_{p^{[d/2]}}$ have a limiting distribution. The first few moments of those coefficients uniquely determine the distributions. For example, for L-functions of degree $d=2$ and weight $w\ge 1$ with rational coefficients there are three possibilities: $$ \langle |a_p|^2 \rangle = 1, \ \ \ \ \ \langle |a_p|^4 \rangle = 2; $$ and $$ \langle |a_p|^2 \rangle = 1, \ \ \ \ \ \langle |a_p|^4 \rangle = 3; $$ and $$ \langle |a_p|^2 \rangle = 2, \ \ \ \ \ \langle |a_p|^4 \rangle = 6. $$ The first possibility corresponds to what was classically known as "Sato-Tate", while the second two are commonly called "CM". For $d=2$ and $w=0$ there are several more options.

Serre uses "Sato-Tate group" for the general case. See his excellent book "Lectures on $N_X(p)$" [MR:2920749].

As the degree grows there are more possibilities: 55 for L-functions of degree $d=4$ and weight $w=1$ with rational coefficients, of which 52 arise for abelian surfaces (in fact, for Jacobians of hyperelliptic curves); this follows from the classification of Fite-Kedlaya-Rotger-Sutherland [10.1112/S0010437X12000279].

Note: The first few coefficient moments determine the distribution because there is a compact matrix group whose eigenvalues have the same distribution as the Satake parameters. That group is the Sato-Tate group of the $L$-function. All of this is conjectural for $d>2$.

Note: The Sato-Tate group determines which operations on the $L$-function ($sym^n$, for example) have poles.