2.4. p-adic L-functions: a first idea
The complex zeta function is an analytic map \zeta:\C\to\C that is rational at
negative integers. Since \Z sits inside both \C and \Zp\subseteq\Cp, it is
natural to look for a p-adic analytic function
\zeta_p:\Zp\longrightarrow\Cp
agreeing with \zeta at negative integers in the sense that, for some explicit
factor (*),
\zeta_p(1-n) = (*)\cdot\zeta(1-n).
One then says \zeta_p p-adically interpolates the special values of \zeta,
and ideally these properties characterise \zeta_p uniquely.
In fact there is no single analytic function on \Zp interpolating all the
special values; the obstruction is the Teichmüller decomposition of \Zpx, and a
clean way to organise the problem is the idelic viewpoint of Tate (and,
independently, Iwasawa), which packages all Dirichlet L-functions — the Riemann
zeta function included — into a single object.
The following hold.
(i) Dirichlet characters are in natural bijection with continuous characters
\chi:\prod_{\ell}\Z_\ell^{\times}\to\C^{\times}, the source carrying the product
of the \ell-adic topologies.
(ii) There is an identification \C\cong\Homc(\R_{>0},\C^{\times}) sending s
to x\mapsto x^s.
Consequently each pair (\chi,s) corresponds to the unique continuous character
\kappa_{\chi,s}:\R_{>0}\times\prod_{\ell}\Z_\ell^{\times}\to\C^{\times},
\qquad (x,y)\mapsto x^s\chi(y),
and every continuous character of this group has this form.
(i) A Dirichlet character \chi:(\Z/N\Z)^{\times}\to\C^{\times} induces a
character of \prod_\ell\Z_\ell^{\times}: when N=\ell^n we use
(\Z/\ell^n\Z)^{\times}\cong\Z_\ell^{\times}/(1+\ell^n\Z_\ell) to inflate \chi
to \Z_\ell^{\times}, and the general case follows by the Chinese remainder
theorem. Conversely, any continuous \chi:\prod_\ell\Z_\ell^{\times}\to\C^{\times}
is trivial on some neighbourhood
U_N=\set{x : x\equiv 1\ (\mathrm{mod}\ N)} of 1: its image lies in
\set{z\in\C : \abs{z-1}<1}, whose only compact subgroup is \set{1}. Thus
\chi factors through the finite quotient
(\prod_\ell\Z_\ell^{\times})/U_N=(\Z/N\Z)^{\times}, giving a Dirichlet
character. The two constructions are mutually inverse.
(ii) Each x\mapsto x^s is a continuous character of \R_{>0}. Conversely,
taking logarithms reduces the claim to: every continuous additive homomorphism
g:\R\to\C has the form g(x)=x\,g(1). This holds on \Q by additivity and
extends to \R by continuity.
The ideles of \Q are the restricted product
\A^{\times} := \R^{\times}\times{\prod_{\ell}}'\,\Q_\ell^{\times}
= \set{(x_\R,x_2,x_3,\dots) : x_\ell\in\Z_\ell^{\times}\text{ for almost all }\ell},
a topological ring with the restricted-product topology (a basis of neighbourhoods
U\times\prod_\ell U_\ell with U\subseteq\R^{\times},
U_\ell\subseteq\Q_\ell^{\times} open and U_\ell=\Z_\ell^{\times} for almost
all \ell). The units \Q^{\times} embed diagonally, x\mapsto(x,x,\dots), and
\Q^{\times}\backslash\A^{\times} is the idele class group of \Q.
- No associated Lean code or declarations.
(Strong approximation.) There is a topological isomorphism
\Q^{\times}\backslash\A^{\times}\cong\R_{>0}\times\prod_{\ell}\Z_\ell^{\times}.
Hence every continuous character \Q^{\times}\backslash\A^{\times}\to\C^{\times}
is of the form \kappa_{\chi,s} for a Dirichlet character \chi and s\in\C.
This rests on Definition 2.4.2 and Proposition 2.4.1.
The isomorphism is the idelic strong approximation theorem for \Q: every idele
class has a unique representative whose finite components lie in
\prod_\ell\Z_\ell^{\times} and whose archimedean component is positive,
obtained by clearing denominators against the diagonal copy of \Q^{\times} and
absorbing the sign. Combined with the classification of characters in
Proposition 2.4.1, every continuous character of the idele class
group is \kappa_{\chi,s}.
Through the identification \C\cong\Homc(\R_{>0},\C^{\times}) one may regard
\zeta as the function [x\mapsto x^s]\mapsto\zeta(s). More strikingly, by
strong approximation all Dirichlet L-functions are values of the single
function
L:\Homc(\Q^{\times}\backslash\A^{\times},\C^{\times})\to\C,\qquad
\kappa_{\chi,s}\mapsto L(\chi,s).
In Tate's framework L integrates \kappa_{\chi,s} against the Haar measure on
the idele class group; this measure-theoretic viewpoint yields analytic
continuation and functional equations, with the \Gamma-factors and powers of
2\pi i appearing as the Euler factor at the archimedean place.