2.5. p-adic L-functions via measures
To p-adicise the idelic picture one replaces \C by \Cp and studies
\Homc(\Q^{\times}\backslash\A^{\times},\Cp^{\times}). Since \R_{>0} is
connected and \Cp is totally disconnected, any such character is trivial on
\R_{>0}; topological arguments show its restriction to
\prod_{\ell\neq p}\Z_\ell^{\times} factors through a finite quotient, hence comes
from a Dirichlet character of conductor prime to p. The interesting part is the
restriction to \Zpx, i.e. \Homc(\Zpx,\Cp^{\times}). We therefore seek a
p-adic analytic function
\zeta_p:\Homc(\Zpx,\Cp^{\times})\to\Cp
that sees the special values of \zeta in the sense that, for an explicit factor
(*),
\zeta_p(x\mapsto x^k) = (*)\cdot\zeta(1-k),\qquad k\geq 1.
The right notion of "p-adic analytic" object here is a p-adic measure
(or pseudo-measure) on \Zpx, developed in the chapter
on measures. With that language in hand, the construction culminates in the
following theorem, the goal of Part I.
(Kubota–Leopoldt, Iwasawa.) There exists a unique pseudo-measure \zeta_p on
\Zpx such that, for all integers k>0,
\int_{\Zpx} x^k\cdot\zeta_p \;=\; \zeta_p(x\mapsto x^k) \;=\; \big(1-p^{k-1}\big)\,\zeta(1-k).
This is the object constructed in the chapter on \zeta_p as the
Definition 4.3.2, and rests on Corollary 2.3.4,
Definition 3.2.2, Definition 3.6.1 and the
Theorem 5.1.1.
This is a forward reference to the central construction of these notes; the full
argument occupies the chapters on the construction of \zeta_p and its
interpolation. The spine is as follows. The special values
\zeta(1-k)=-B_k/k lie in \Q, hence in \Cp, by
Corollary 2.3.4. The Euler factor 1-p^{k-1} is exactly the
inverse of the local factor (1-\ell^{-s})^{-1} of Definition 2.1.1 at
\ell=p, evaluated at s=1-k; thus the theorem p-adically interpolates the
p-deprived zeta function. One builds explicit measures \mu_a on \Zp,
restricts them to \Zpx, and rescales away the auxiliary parameter a to
produce \zeta_p; the interpolation formula is then verified by integrating the
monomials x^k against the Mahler/Mellin description of the measure. Uniqueness
holds because the monomials x^k, k>0, are dense enough in the space of
continuous characters to pin down a pseudo-measure.
The factor 1-p^{k-1} is the inverse of the Euler factor at p of
\zeta(s)=\prod_\ell(1-\ell^{-s})^{-1} at s=1-k. Although the Euler product
diverges at s=1-k, Theorem 2.5.1 morally says that after
removing the factor at p the Riemann zeta function interpolates p-adically.
This p-stabilisation is a general feature of the theory.
Although \zeta_p is built using only values of \zeta — with no reference to
Dirichlet characters — the measure-theoretic viewpoint gives far more: the same
pseudo-measure simultaneously interpolates all Dirichlet L-functions of
p-power conductor.
Let \chi be a Dirichlet character of conductor p^n, n\geq 0, viewed as a
locally constant character on \Zpx. Then for all k>0,
\int_{\Zpx}\chi(x)\,x^k\cdot\zeta_p \;=\; \big(1-\chi(p)p^{k-1}\big)\,L(\chi,1-k).
This rests on Theorem 2.5.1,
Definition 2.1.3 and the Theorem 5.1.1.
This is the character-twisted interpolation property, proved in the interpolation
chapter. The point is that integrating against the locally constant character
\chi only modifies the measure \zeta_p on the cosets of
1+p^n\Zp in \Zpx; reorganising the resulting finite sum of partial integrals
reconstitutes the Dirichlet series L(\chi,1-k), while the local factor at p
becomes 1-\chi(p)p^{k-1} (with \chi(p)=0 when p\mid conductor). Since
\zeta_p was defined using untwisted values only, that it also encodes twisted
values is genuinely surprising.
Let D>1 be coprime to p and let \eta be a primitive Dirichlet character of
conductor D. There is a unique measure \zeta_\eta on \Zpx such that, for
every primitive Dirichlet character \chi of conductor p^n, n\geq 0, and all
k>0,
\int_{\Zpx}\chi(x)\,x^k\cdot\zeta_\eta \;=\; \big(1-\chi\eta(p)p^{k-1}\big)\,L(\chi\eta,1-k).
This rests on Theorem 2.5.2 and
Definition 2.1.3.
The tame character \eta of conductor prime to p is incorporated by an
auxiliary twist of the construction of \zeta_p, again realised as a p-adic
measure on \Zpx. As \eta ranges over characters of conductor prime to p
the measures \zeta_\eta are compatible under the natural maps
(\Z/E\Z)^{\times\wedge}\to(\Z/D\Z)^{\times\wedge} for E\mid D, so they assemble
into a single function on
\Homc(\Zpx,\Cp^{\times})\times(\prod_{\ell\neq p}\Z_\ell^{\times})^{\wedge}
=\Homc(\Q^{\times}\backslash\A^{\times},\Cp^{\times}). This is precisely the
p-adic counterpart of the idelic function
L:\kappa_{\chi,s}\mapsto L(\chi,s): a measure on the idele class group of \Q.
- No associated Lean code or declarations.
(Generalised Kummer congruences.) Let \eta be a Dirichlet character of
conductor prime to p. If k\equiv j\ (\mathrm{mod}\ p^{m-1}(p-1)) with k,j>0,
then the p-stabilised special values satisfy
\big(1-\eta(p)p^{k-1}\big)L(\eta,1-k) \equiv \big(1-\eta(p)p^{j-1}\big)L(\eta,1-j)\ \ (\mathrm{mod}\ p^m).
For \eta=1 these are the classical Kummer congruences for the Riemann zeta
function. This rests on Corollary 2.3.4 and
Theorem 2.5.3.
The congruence k\equiv j\ (\mathrm{mod}\ p^{m-1}(p-1)) forces
x^k\equiv x^j\ (\mathrm{mod}\ p^m) for every x\in\Zpx, because the finite
quotient \Zpx/(1+p^m\Zp)\cong(\Z/p^m\Z)^{\times} has order p^{m-1}(p-1), so
raising to the exponent kills the difference k-j. Hence the two locally
constant functions x\mapsto\eta(x)x^k and x\mapsto\eta(x)x^j are congruent
modulo p^m pointwise on \Zpx. Integrating both against the measure
\zeta_\eta of Theorem 2.5.3 — whose values are
p-adic integers up to the bounded denominators of a measure — and using the
interpolation formula
\int_{\Zpx}\eta(x)x^k\cdot\zeta_\eta=(1-\eta(p)p^{k-1})L(\eta,1-k) yields the
displayed congruence. These congruences underlie Kummer's classification of
irregular primes and were the historical motivation for
Theorem 2.5.1; they give the complementary view of p-adic
L-functions as analytic objects packaging systematic congruences between
L-values.