Rubin's conjecture on local units in the anticyclotomic tower at inert primes
Let \(p \geq 5\) be a prime. The authors prove a conjecture of \textit{K. Rubin} [Invent. Math. 88, 405--422 (1987; Zbl 0623.14006)] on the structure of local units along the anticyclotomic \(\mathbb{Z}_p\)-extension of the unramified quadratic extension \(\Phi\) of \(\mathbb{Q}_p\). \par We briefly sketch Rubin's conjecture. Fix a Lubin-Tate formal group \(\mathcal{F}\) over the ring of integers \(\mathcal{O}\) in \(\Phi\) for the uniformizer \(\pi := -p\). Let \(\Phi_n\) be the field obtained from \(\Phi\) by adjoining all \(\pi^{n+1}\)-torsion points of \(\mathcal{F}\) and set \(\Phi_{\infty} := \bigcup_n \Phi_n\). Then there is a natural isomorphism \(\kappa:\mathrm{Gal}(\Phi_{\infty}/\Phi) \simeq \mathcal{O}^{\times}\). Moreover, one has a decomposition \[ \mathrm{Gal}(\Phi_{\infty}/\Phi_0) \simeq G^+ \times G^-, \] where both \(G^{\pm}\) are isomorphic to \(\mathbb{Z}_p\) and \(\mathrm{Gal}(\Phi/\mathbb{Q}_p)\) acts upon \(G^+\) and \(G^-\) by \(+1\) and \(-1\), respectively. We may identify \(G^+\) and \(G^-\) with the Galois groups of the cyclotomic and the anticyclotomic \(\mathbb{Z}_p\)-extension of \(\Phi\), respectively. \par Let \(U_n\) be the group of principal units in \(\Phi_n\) and consider the inverse limit \(\varprojlim_n U_n \otimes_{\mathbb{Z}_p} \mathcal{O}\) with respect to the norm maps. We denote the part of this limit upon which \(\mathrm{Gal}(\Phi_0/\Phi)\) acts via the Teichmüller character by \(U_{\infty}\). Then \(U_{\infty}\) is free of rank \(2\) over the Iwasawa algebra \(\mathcal{O}[[\mathrm{Gal}(\Phi_{\infty}/\Phi_0)]]\). The main object of study is the quotient \[ V^{\ast}_{\infty} := U_{\infty}^{\ast} / (\sigma-1), \] where \(U_{\infty}^{\ast}\) is \(U_{\infty}\) with the Galois action twisted by \(\kappa^{-1}\) and \(\sigma\) is a topological generator of \(G^+\). Then \(V^{\ast}_{\infty}\) is a free \(\Lambda := \mathcal{O}G^-\)-module of rank \(2\). Rubin defines two subspaces \(V^{\ast, \pm}_{\infty}\) of \(V^{\ast}_{\infty}\) and conjectures that one has a decomposition \[ V^{\ast}_{\infty} = V^{\ast,+}_{\infty} \oplus V^{\ast,-}_{\infty}. \] He showed that both subspaces are free of rank \(1\) and that their intersection is trivial. \par The strategy of proof of Rubin's conjecture is now as follows. Consider the Coates-Wiles derivative \[ \delta: V^{\ast}_{\infty} / V^{\ast,-}_{\infty} \rightarrow \mathcal{O}. \] The authors prove that (i) there is \(\xi \in V^{\ast,+}_{\infty}\) such that \(\delta(\xi) \in \mathcal{O}^{\times}\) and (ii) there is an isomorphism \(V^{\ast}_{\infty}/V^{\ast,-}_{\infty} \simeq \Lambda\). Under this identification, \(\xi\) is not contained in the maximal ideal of \(\Lambda\) (as \(\delta(\xi) \in p \mathcal{O}\) otherwise) and hence is a generator of the quotient \(V^{\ast}_{\infty} / V^{\ast,-}_{\infty}\). Rubin's conjecture follows. \par For (i) the authors construct a certain auxiliary imaginary quadratic field \(K\) and an elliptic curve \(E\) over its Hilbert class field with complex multiplication by \(\mathcal{O}_K\). By a criterion of Rubin it would be sufficient to construct such an \(E\) with good supersingular reduction at \(p\) whose central \(L\)-value is \(p\)-divisible. The authors generalize this approach to allow more general \(L\)-values (they consider twists by certain Hecke characters). The proof of (i) then crucially relies on work of \textit{T. Finis} [Ann. Math. (2) 163, No. 3, 767--807 (2006; Zbl 1111.11047)]. For (ii) the authors make use of the theory of quasi-canonical lifts of \textit{B.H. Gross} [Invent. Math. 84, 321--326 (1986; Zbl 0597.14044)] to construct an optimal system of local points of the formal group. This is the main difficulty of the present work. \par Finally, we note that the main result of this article makes the work of \textit{A. Agboola} and \textit{B. Howard} [Math. Res. Lett. 12, No. 5--6, 611--621 (2005; Zbl 1130.11058)] on a variant of the Iwasawa main conjecture unconditional.
scientific article; zbMATH DE number 7441739
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