9.6. Generalisations: Kummer sequence, Euler systems and big logarithms
This final section sketches the conjectural framework — due to Perrin-Riou — that
generalises the Coleman map to arbitrary p-adic Galois representations. It is
context for Part II and may be skipped on a first reading. For a number field F
write \sG_F for its absolute Galois group and H^1(F, A) := H^1(\sG_F, A) for
continuous Galois cohomology.
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Taking the long exact cohomology sequence of the Kummer sequence
0 \to \mu_{p^m} \to \mathbf{G}_{\mathrm m} \xrightarrow{x\mapsto x^{p^m}} \mathbf{G}_{\mathrm m} \to 0
over \Qbar, and passing to the inverse limit over m, yields the Kummer
isomorphism
\delta : F^{\times} \otimes \Zp \xrightarrow{\ \sim\ } H^1(F, \Zp(1)).
For each m the long exact sequence reads
0 \to \mu_{p^m}(F) \to F^\times \xrightarrow{x^{p^m}} F^\times \to H^1(F,\mu_{p^m}) \to H^1(F, \Qbar^\times),
and H^1(F,\Qbar^\times) = 0 by Hilbert 90. Thus
F^\times/(F^\times)^{p^m} \xrightarrow{\sim} H^1(F,\mu_{p^m}). Explicitly, given
a \in F^\times pick b with b^{p^m} = a; then \sigma \mapsto \sigma(b)/b
is a 1-cocycle, a coboundary iff a is a p^mth power, giving the level-m
isomorphism. The inverse limit over m (exact, as the systems are finite) assembles
these into \delta : F^\times \otimes \Zp \xrightarrow{\sim} H^1(F,\Zp(1)).
Applied to F = \Q(\mu_m) with m = Dp^n, the elements
\mathbf c_m := (\xi_m^{-1}-1)/(\xi_m - 1) (generalising c_n(-1)) map to classes
\mathbf z_m := \delta(\mathbf c_m) \in H^1(\Q(\mu_m),\Zp(1)) satisfying, under
corestriction, \cores(\mathbf z_{m\ell}) = \mathbf z_m if \ell \mid m and
(1 - \Frob_\ell^{-1})\mathbf z_m if \ell \nmid m — using that \Frob_\ell
acts on \Zp(1) by \ell, and noting 1 - \ell^{-1} is the Euler factor at
\ell of \zeta(s) at s=1. This is the prototype of an Euler system.
Let \Sigma be a finite set of primes containing p, let
V \in \mathrm{Rep}_L\,\sG_\Q be a global p-adic Galois representation
unramified outside \Sigma, and let T \subseteq V be a \sG_\Q-stable
\cO_L-lattice. An Euler system for (V,T,\Sigma) is a collection of classes
\mathbf z_m \in H^1(\Q(\mu_m), T), with m = p^n m' and m' a square-free
product of primes outside \Sigma, satisfying
\cores_{\Q(\mu_{m\ell})/\Q(\mu_m)}(\mathbf z_{m\ell}) = \begin{cases} \mathbf z_m & \ell = p,\\ P_\ell(V^*(1), \sigma_\ell^{-1})\,\mathbf z_m & \ell \neq p, \end{cases}
where P_\ell(V^*(1), X) = \det(1 - \Frob_\ell^{-1}X \mid V^*(1)^{I_\ell}) is the
Euler factor of L(V^*(1), s) at \ell and \sigma_\ell is the image of
\Frob_\ell in \Gal(\Q(\mu_m)/\Q).
The cyclotomic units Definition 9.3.1 form an Euler system
Definition 9.6.2 for \Zp(1), central to Rubin's proof of the Main
Conjecture.
Replacing \Qbar by \Qpbar and F by K_n in
Proposition 9.6.1 gives
K_n^\times \otimes \Zp \cong H^1(K_n,\Zp(1)), intertwining norm with
corestriction. Taking the inverse limit, the Iwasawa cohomology is
H^1_{\Iw}(\Qp, \Qp(1)) := \varprojlim_{n\ge 1} H^1(K_n, \Zp(1)) \otimes_{\Zp} \Qp,
and more generally, for V \in \mathrm{Rep}_L\,\sG_{\Qp} with stable lattice T,
H^1_{\Iw}(\Qp, V) := \varprojlim_n H^1(K_n, T)\otimes_{\cO_L} L (limit over
corestriction). The inclusion \sU_n \subset K_n^\times induces
\kappa : \sU_\infty \to \varprojlim_n H^1(K_n,\Zp(1)).
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There is a map \Col' : H^1_{\Iw}(\Qp,\Qp(1)) \to \cM(\Zpx,\Qp) to the space of
\Qp-valued measures, factoring the Coleman map: \Col' \circ \kappa = \Col.
More generally, for crystalline V, Perrin-Riou's big logarithm map
\Log_V : H^1_{\Iw}(\Qp, V) \to \cD^{\mathrm{la}}(\Zpx, L)
into locally analytic distributions specialises to \Col' at V = \Qp(1). Thus a
p-adic L-function for V arises from an Euler system by localising at p
and applying \Log_V:
\{\text{Euler systems}\} \xrightarrow{\ \loc_p\ } H^1_{\Iw}(\Qp, V) \xrightarrow{\ \Log_V\ } \{p\text{-adic }L\text{-functions}\}.
The factorisation \Col' \circ \kappa = \Col is the diagram defining \Col' on
the image of \kappa Definition 9.6.3 and extending by the Iwasawa
algebra structure; combined with Proposition 9.3.3 it shows
\zeta_p is the value of \Col' at the cyclotomic Iwasawa class (up to dividing
by \theta_a, which introduces the pole), recovering
Theorem 9.5.2. The existence and interpolation of \Log_V for
crystalline V, expressed through Bloch–Kato's exponential and dual-exponential
maps, is Perrin-Riou's theory; we record only that it specialises to \Col' at
V = \Qp(1). This splits the construction of p-adic L-functions into a global
problem (producing an Euler system, e.g. via Definition 9.6.2) and a
purely local one (the big logarithm).