Iwasawa theory and the Eisenstein ideal
Let \(p\) be an odd prime, \(F = {\mathbb Q}(\mu_p)\), \({\mathcal G}\) = the Galois group over \(F\) of the maximal pro-\(p\) unramified outside \(p\) extension of \(F.\) In their paper [Duke Math. J. 120, 269--310 (2003; Zbl 1047.11106)], \textit{W. G. McCallum} and \textit{R. T. Sharifi} studied the cup product \(H^1({\mathcal G}, \mu_p) \times H^1({\mathcal G}, \mu_p) \to A_F \otimes \mu_p\), where \(A_F\) is the \(p\)-class group of \(F\). In particular, they conjectured that the pairing induced by this cup product on the \(p\)-units \(\xi_F\) of \(F,\) \((\ldotp, \ldotp): \xi_F \times \xi_F \to A_F \otimes \mu_p,\) is surjective on the minus part. In this paper, the second named author shows that the conjecture holds for \(p < 10^3.\) The result itself is less interesting than the method of proof, which reveals an intriguing relationship between the structure of certain Iwasawa modules over Kummer extensions and the structure of ordinary Hecke algebras of modular forms localized at the Eisenstein ideal, in the same streamline as Ribet's proof of the converse of Herbrand's theorem (1976), Mazur-Wiles' proof of the Main Conjecture over \({\mathbb Q}\) (1984), and Ohta's revisitation of Mazur-Wiles using Hida theory (2000). More specifically, fix an even integer \(k < p\) such that \(p \mid B_k,\;p \nmid B_{p+1-k}\) and, for simplicity of this description, \(p^2 \nmid B_k.\) The author studies the action of \(G_{\mathbb Q}\) on an Eisenstein component \(\mathfrak X\) of a particular inverse limit of cohomology groups of modular curves previously considered by \textit{M. Ohta} [J. Reine Angew. Math. 585, 141--172 (2005; Zbl 1081.11035)]. This \(\mathfrak X\) is a free module of rank 2 over \({\mathfrak H},\) the localization of Hida's ordinary cuspidal Hecke algebra at the maximal ideal containing the Eisenstein ideal \({\mathcal J}\) with character \(\omega^k.\) Let \(f : G_{\mathbb Q} \to \Aut_{\mathfrak H}(\mathfrak X), \;\sigma \mapsto \left(\begin{smallmatrix} a(\sigma) &b(\sigma)\\ c(\sigma) &d(\sigma)\end{smallmatrix}\right),\) be the Galois representation on \(\mathfrak X.\) It was shown by Ohta that, under natural isomorphisms, the ideals \(B\) and \(C\) generated by the images of \(b\) and \(c\) verify \(B = {\mathcal J}\) and \(C = {\mathfrak H},\) and moreover \(X_K (\omega^{1-k}) \simeq B/{\mathcal J} B\) as modules over \(\widetilde\Gamma= \text{Gal} (K/{\mathbb Q})\) [\textit{M. Ohta}, Math. Ann. 318, 557--583 (2000; Zbl 0967.11016)]. Here \(K\) is the subfield of \({\mathbb Q} (\mu_{p^\infty})\) cut out by the kernel of \(\omega^k\) and \(X_K\) is the usual unramified Iwasawa module over \(K.\) Composing \(\rho\) with the projection of \(\Im \rho\) to matrices with entries \(\alpha \in ({\mathfrak H}/{\mathcal J})^\times, \beta \in {\mathcal J}/{\mathcal J}^2,\) \( \gamma \in {\mathfrak H}/{\mathcal J}\), \(\delta \in ({\mathfrak H}/{\mathcal J}^2)^\times\), the author shows that the kernel of the Galois representation thus obtained cuts out a Heisenberg extension \(M/K\) such that \(M^{ab} = HL,\) where \(\text{Gal} (H/K) \simeq X_K (\omega^{1-k})\) and \(\text{Gal} (HL/L) \simeq (X_L/I_G X_L) (\omega^{1-k}),\) \(G = \text{Gal} (L/K)\); moreover, \(\text{Gal} (M/HL) \simeq (I_G X_L/I_G^2 X_L)_{\widetilde \Gamma} \simeq {\mathcal J}/{\mathcal J}^2\), and under this last isomorphism, the Frobenius element on the Galois side corresponds to \(U_p-1\) on the Hecke side. It follows that the corresponding subquotient of the unramified, \(p\)-split Iwasawa module over \(L\) is isomorphic to \({\mathcal J}/{\mathcal J}^2\) modulo \(\langle U_p-1 \rangle\). The structure of these latter subquotients can be seen to relate to the \(\omega^{1-k}\)-eigenspace of \(X_K\) modulo the submodule generated by an inverse limit of cup products up the cyclotomic tower. Then \((p, \ldotp)\) is surjective if and only if \(U_p-1\) generates \({\mathcal J}\) for each \(k,\) which can be checked numerically for \(p < 10^3\). Note that the author does not conjecture that \((p, \ldotp)\) itself is always surjective, nor that \(U_p -1\) always generates \({\mathcal J}\). In a recent work (preprint, 2007), he discusses a conjectural relationship between \((\eta_i, \eta_{k-i})\) for special cyclotomic \(p\)-units \(\eta_i\) (\(i\) odd) and \(L_p (f, \omega^{i-1}, 1),\) where \(f\) is a certain cuspidal eigenform (considered by Ribet) such that the kernel of the modular representation \(\rho_f\) cuts out the analog at finite level of the field \(H\) above.
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