Hecke algebras associated to \(\Lambda\)-adic modular forms (Q2261608)

The author studies whether the Eisenstein component of the \(p\)-adic Hecke algebra associated to modular forms is Gorenstein, and how this relates to the theory of cyclotomic fields via a conjecture of \textit{R. Sharifi} [Ann. Math. (2) 173, No. 1, 251--300 (2011; Zbl 1248.11085)], partially proved by \textit{T. Fukaya} and \textit{K. Kato} [``On conjectures of Sharifi'' (submitted)]. We first need quite a few notations. Let \(p\geq 5\) be a prime and \(N\) an integer such that \(p\nmid N\) and \(p\nmid\varphi(N)\). Let \(\theta: (\mathbb{Z}/Np\mathbb{Z})^\times\to \overline{\mathbb{Q}_p}^\times\) be an even character and let \(\chi=\omega^{-1}\theta\), \(\omega\) denoting the Teichmüller character. Suppose that, as by Sharifi [loc. cit.], \(\theta\) satifies certain technical conditions which we do not recall. On the modular side, let \(H'= \text{proj.lim\,} H^1(\overline X_1(Np^r), \mathbb{Z}_p)^{\text{ord}}_\theta\), \(\widetilde H'= \text{proj.lim\,} H^1(\overline Y_1(Np^r), \mathbb{Z}_p)^{\text{ord}}_\theta\), where ord denotes the ordinary part for the dual Hecke operator \(T^*(p)\), and the subscript refers to the eigenspace for the diamond operators. Let \({\mathfrak h}'\) (resp. \({\mathfrak H}'\)) be the algebra of dual Hecke operators acting on \(H'\) (resp. \(\widetilde H'\)), and \(I\) (resp. \({\mathcal I}\)) the Eisenstein ideal of \({\mathfrak h}'\) (resp. \({\mathfrak H}'\)). Consider the Eisenstein component \({\mathfrak H}\), the localization of \({\mathfrak H}'\) at the unique maximal ideal containing \({\mathfrak I}\), and analogously the Eisenstein component \({\mathfrak h}\). Let \(\widetilde H= \widetilde H' \otimes_{{\mathfrak H}'}{\mathfrak H}\) and \(H= H'\otimes_{{\mathfrak h}'}{\mathfrak h}\). On the Iwasawa theoretic side, let \(\mathbb{Q}_\infty= \mathbb{Q}(\zeta_{Np^\infty})\), \(M=\) the maximal Abelian pro-\(p\)-extension of \(\mathbb{Q}_\infty\) unramified outside \(Np\), \(L=\) the maximal Abelian pro-\(p\)-extension of \(\mathbb{Q}_\infty\) unramified everywhere, \({\mathfrak X}=\text{Gal}(M/\mathbb{Q}_\infty)\), \(X= \text{Gal}(L/\mathbb{Q}_\infty)\). Let \(\mathbb{Z}^x_{p,N}= \mathbb{Z}^x_p\times(\mathbb{Z}/N\mathbb{Z})^x\) and consider the Iwasawa algebra \(\Lambda= \mathbb{Z}_p[[\mathbb{Z}^\times_{p, N}]]_\theta\cong \mathbb{Z}_p[[\text{Gal}(\mathbb{Q}_\infty/\mathbb{Q})]]_\theta\), where the subscript refers to the eigenspace for the Galois action. Denote by \((.)^\pm\) the eigenspaces for complex conjugation. It has been shown by \textit{C. M. Skinner} and \textit{A. J. Wiles} [Proc. Natl. Acad. Sci. USA 94, No. 20, 10520--10527 (1997; Zbl 0924.11044)] and by \textit{M. Ohta} [J. Math. Soc. Japan 59, No. 4, 913--951 (2007; Zbl 1187.11014)], using different methods, that \({\mathfrak H}\) is Gorenstein if \({\mathfrak X}^+_\theta= 0\). But note that the hypothesis \({\mathfrak X}^+0_\theta= 0\) implies a considerable simplification on the Galois side, namely the freeness of the Galois group of the maximal pro-\(p\)-extension uramified outside \(Np\) of the totally real real field cut out by \(\text{Ker}\,\theta\). Replacing \({\mathfrak X}^+_\theta\) by its quotient \(X^+_\theta\), the author shows the following necessary and sufficient condition: Suppose that \(X^-_\chi\neq 0\) (if \(X^-_\chi=0\), then \({\mathfrak H}=\Lambda\)). Then: (1) If \({\mathfrak H}\) is Gorenstein, \(X^+_\theta= 0\); (2) If Sharifi's conjecture is true, \(X^+_\theta= 0\) implies that \({\mathfrak H}\) is Gorenstein. Recall that the conjecture of Sharifi [loc. cit.] states the bijectivity of a certain homomorphism \({\mathfrak Y}: X^-+\chi\to\text{Hom}_{{\mathfrak h}}({\mathcal Q},{\mathcal P})\), with \({\mathcal P}= H^-/IH^-\) and \({\mathcal Q}=(H/IH)/{\mathcal P}\cong(\Lambda/\xi)^{\#}\), where \(\xi\) is a characteristic power series of \(X^-_\chi\) and \((.)^{\#}\) means that the Galois action is inverted. \(NB : X^-_\chi\) needs to be twisted one time in order to become a \(\Lambda\)-module; the author's notation assumes this implicitly. A corollary is that \({\mathfrak H}\) is not always Gorenstein, counterexamples being provided by the case where both \(X^-_\chi\) and \(X^+_\theta\) are non-null. Recall that if \(N=1\), Vandiver's conjecture implies \(X^+_\theta= 0\). Note also that the author has formulated a ``weak Gorenstein property'' for \({\mathfrak H}\) based on whether certain localizations of \({\mathfrak H}\) are Gorenstein, and has proved that it holds if and only if certain weak forms of Sharifi's conjecture and of Greenberg's conjecture simultaneously hold [Algebra and Number Theory 9, No. 1, 53--75 (2015)]. The proof of the main theorem of the present paper starts from an idea of Fukaya-Kato [loc. cit.]. Let \({\mathcal R}=({\mathfrak h}\otimes_{{\mathfrak H}}/H\cong\Lambda/\xi(- 1)\). The author constructs a commutative diagram \[ \begin{tikzcd} {\mathfrak X}^-_{\chi^-1}\otimes X^-_\chi \ar[r, "\Theta\otimes{\mathcal Y}"]\ar[d,"\nu\otimes{\mathcal Y}" '] & \Hom_{\mathfrak h}({\mathcal R},{\mathcal Q})\otimes\Hom_{{\mathfrak h}}({\mathcal Q},{\mathcal P})\ar[d] \\ (\Lambda/\xi)^{\#}\otimes\Hom_{\mathfrak h}(\mathcal Q,{\mathcal P}) \ar[r]&{\mathcal P}(1)\end{tikzcd} \] (we don't recall the definitions of \(\Theta\) and \(\nu\)). He shows first that \({\mathfrak H}\) is Gorenstein if and only if the clockwise map \({\mathcal X}^-_{\chi^{-1}}\otimes X^-_\chi\to {\mathcal P}(1)\) is surjective. It is known that if \({\mathfrak H}\) is Gorenstein, then \({\mathcal Y}\) is an isomorphism. Since \(X^-_\chi\neq 0\), we get that if \({\mathfrak H}\) is Gorenstein, then \(\nu\) is surjective. Using Iwasawa adjoints, the author finally shows that \(\nu\) is surjective if and only if \(X^+_\theta= 0\).