12.5. Generalisations: Selmer groups and the Iwasawa–Greenberg Main Conjecture
We conclude with a sketch of Greenberg's formulation of a Main Conjecture for
general Galois representations, which recovers the classical statement for the
trivial representation. The objects on the algebraic side are Selmer groups. We
work over \cF = F_\infty^+ = \Q(\mu_{p^\infty})^+, with coefficients in a fixed
finite extension L/\Qp.
- No associated Lean code or declarations.
Let M be a topological \cO_L-module with a continuous \cO_L-linear action
of \GG_\cF = \Gal(\overline{\cF}/\cF), unramified outside a finite set of
places. A Selmer structure \cL = (\cL_v)_v for M is a choice of subspace
\cL_v \subseteq H^1(\cF_v, M) for every finite place v of \cF, with
\cL_v = H^1_{\mathrm{ur}}(\cF_v, M) for almost all v, where the unramified
subgroup is H^1_{\mathrm{ur}}(\cF_v, M) = \ker\big(H^1(\cF_v, M) \to H^1(I_{\cF_v}, M)\big)
with I_{\cF_v} the inertia group. The associated Selmer group is
H^1_{\cL}(\cF, M) = \ker\Big( H^1(\cF, M) \to \bigoplus_v H^1(\cF_v, M)/\cL_v \Big),
the sum over all finite places v.
We take T a finite free \Zp-module with a \GG_\Q-action attached to some
arithmetic object, set V = T\otimes_{\Zp}\Qp and W = V/T \cong V\otimes(\Qp/\Zp),
and take M = W. Away from p we always impose the unramified local condition;
at p there are two standard prescriptions.
With \cF = \Q(\mu_{p^\infty})^+ and v_p the unique place of \cF above
p:
(Greenberg) Assume V is p-ordinary: there is a saturated finite
\GG_\Q-stable filtration \Fil^\bullet V on which the inertia group
I_{\Qp} acts on \Fil^i/\Fil^{i+1} through the i-th power of the cyclotomic
character. Then
\cL_{v_p}^{\Gr} = \ker\Big( H^1(\cF_{v_p}, W) \to H^1(I_{v_p}, W) \to H^1(I_{v_p}, W/\Fil^1 W) \Big),
where \Fil^1 W is the image of \Fil^1 V in W.
(Bloch–Kato) Using p-adic Hodge theory and Fontaine's ring \Bcris of
crystalline periods, set H^1_f(\cF_{v_p}, V) = \ker\big(H^1(\cF_{v_p}, V) \to H^1(\cF_{v_p}, V\otimes\Bcris)\big)
and \cL_{v_p}^{\mathrm{BK}} = \Im\big(H^1_f(\cF_{v_p}, V) \to H^1(\cF_{v_p}, W)\big).
This refines the Selmer structure of Definition 12.5.1.
The two prescriptions agree in many fundamental cases. We now reinterpret
\sX_\infty^+ as a Greenberg Selmer group for the cyclotomic twists
W_n = (\Qp/\Zp)(n).
Let n be even and positive. With T_n = \Zp(n), V_n = \Qp(n),
W_n = V_n/T_n and the Greenberg local conditions of Definition 12.5.2,
the Selmer group over \cF = F_\infty^+ is
H^1_{\cL^{\Gr}}(F_\infty, W_n) = \Hom_{\mathrm{cts}}(\sX_\infty^+, W_n),
a twist of the Pontryagin dual of the module \sX_\infty^+ of the Main
Conjecture. This uses Definition 12.2.2 and Definition 12.2.3.
Since \GG_{F_\infty} acts trivially on W_n, cohomology classes are continuous
homomorphisms: H^1(F_\infty, W_n) = \Hom_{\mathrm{cts}}(\GG_{F_\infty}, W_n).
As \Gal(F_\infty/F_\infty^+) = \{1, c\} \cong \{\pm 1\} and p is odd,
inflation–restriction gives
H^1(\cF, W_n) = \Hom_{\mathrm{cts}, \{\pm1\}}(\GG_{F_\infty}, W_n), the
\{\pm1\}-equivariant homomorphisms (the \GG-action being conjugation as in
Definition 12.2.3). Imposing the unramified condition at all v \nmid p
factors such a homomorphism through
\sX_\infty = \Gal(\sM_\infty/F_\infty). At p, the Greenberg filtration
\Fil^i\Qp(n) = \Qp(n) for i \le n and 0 otherwise makes the local
condition \cL_{v_p}^{\Gr} empty when n \ge 1, so no further constraint is
imposed and the class descends to \sX_\infty. Finally c acts on W_n by
(-1)^n, so for even n>0 only the c=+1 part survives:
H^1_{\cL^{\Gr}}(F_\infty, W_n) = \Hom_{\mathrm{cts}}((\sX_\infty)^{c=1}, W_n). The
natural surjection \sX_\infty \twoheadrightarrow \sX_\infty^+ induces
(\sX_\infty)^{c=1} \cong \sX_\infty^+, giving the stated identification with
\Hom_{\mathrm{cts}}(\sX_\infty^+, W_n). For n \le 0 the same analysis adds the
unramified condition at p, replacing \sX_\infty by \sY_\infty.
- No associated Lean code or declarations.
(Iwasawa–Greenberg Main Conjecture, conjectural.) Let V be p-ordinary with
Tate dual V^\vee = \Hom_{\mathrm{cts}}(V, \Qp(1)),
let L_\infty(V,s) be the Gamma factor of V (a zero-free product of translates
of \Gamma(s)), and let r_V be the order of the pole of L_\infty(V,s) at
s=1. Then:
(i) H^1_{\cL^{\Gr}}(\cF, W) has \Lam(\Gamma^+)-corank equal to r_V.
(ii) If r_V = r_{V^\vee} = 0, the characteristic ideal of the Pontryagin dual
of H^1_{\cL^{\Gr}}(\cF, W) equals the ideal generated by the conjectural
p-adic L-function of V (an element of the fraction field of
\Lam(\Gamma^+), predicted by Coates–Perrin-Riou). This generalises
Theorem 12.3.1 and refers to Definition 12.5.1,
Definition 12.5.2 and Definition 12.1.4.
This is a conjecture, recorded here for context; we only indicate why it recovers
the classical case. For even n > 0 the Selmer group is
\Hom_{\mathrm{cts}}(\sX_\infty^+, W_n) by Proposition 12.5.3, whose
Pontryagin dual is \sX_\infty^+(-n), the twist of \sX_\infty^+ by
\chi^{-n}. In this case Greenberg's p-adic L-function is \partial^n\zeta_p,
the n-th twist of Kubota–Leopoldt Definition 4.3.2, so part (ii)
becomes a twist of the Iwasawa Main Conjecture Theorem 12.3.1.
Strictly the case n=0 falls outside Greenberg's hypotheses, since
\Qp^\vee = \Qp(1) forces r_{\Qp^\vee} = 1; but the two statements are
equivalent. More general — even non-commutative — formulations exist in the work
of Kato and of Fukaya–Kato.