Quaternion algebras, Heegner points and the arithmetic of Hida families (Q634806)

The principal goal of this article is to extend the work of \textit{B. Howard} [Invent. Math. 167, No. 1, 91--128 (2007; Zbl 1171.11033)], who made use of (appropriately twisted) Heegner points along a tower of modular curves so as to introduce an Euler system associated to (the central critical twist of) Hida's universal nearly ordinary deformation representation. This work was already extended by \textit{O. Fouquet} [Publ. Math. Univ. Franche-Comté Besançon, Algèbre et Théorie des Nombres 2007--2009, 79--95 (2009; Zbl 1158.11001)] to treat towers of Shimura curves over totally real fields, which are associated to an indefinite quaternion algebra. These constructions have applications (a la Kolyvagin) towards the Hida-theoretic versions of Perrin-Riou's Heegner point main conjectures, see the work of \textit{O. Fouquet} [Compos. Math. 149, No. 3, 356--416 (2013; Zbl 1286.11066)] and the reviewer [Sel. Math., New Ser. 20, No. 3, 787--815 (2014; Zbl 1296.11057)] (Although it is not directly related to the contents of this article, the reviewer felt that one should record here that said conjecture of Perrin-Riou was recently settled by Xin Wan.) The significance of the current article lies in the fact that the authors (at least in part) manage to generalize all the previous work to treat Shimura Curves attached to a definite quaternion algebra. Having said that, we should note that they also present results in the indefinite setting that, in a certain sense of the word, complementing the work of Fouquet [Zbl 1158.11001]. The main difference in the definite set up is that the Shimura curves in question are a finite union of genus zero curves, and the construction of the required CM points require a different idea. The authors obtain these objects utilizing the theory of optimal embeddings. In order to make us of the collection of these objects, the authors are led to a conjectural generalization of the work of \textit{M. Bertolini} and \textit{H. Darmon} [Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)] and their own previous work [\textit{M. Longo} and \textit{S. Vigni}, J. Number Theory 130, No. 1, 128--163 (2010; Zbl 1262.11068)]. This conjecture (which is part of the statement of Conjecture 9.6 in the text) predicts the vanishing of the Bloch-Kato Selmer group attached to an arithmetic point \(\mathfrak{p}\), provided that the divisor the authors obtained out of CM points does not vanish under specialization at \(\mathfrak{p}\). They also formulate a two-variable main conjecture in this set up, much in the spirit of the constructions carried out in [\textit{M. Bertolini} and \textit{H. Darmon}, Invent. Math. 126, No. 3, 413--456 (1996; Zbl 0882.11034); Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)], which should be thought of as an extension of Howard's main conjecture to Shimura curves associated to definite quaternion algebras. The exposition of this informative article is rather clean and the reviewer highly recommends this nice work for a pleasant read.