The universal \(p\)-adic Gross-Zagier formula
The beautiful and important paper under review offers a new viewpoint in the long and rich area of \(p\)-adic interpolation of arithmetic cycles along \(p\)-adic families of modular forms. Specifically, starting with the seminal work of \textit{B. Howard} [Invent. Math. 167, No. 1, 91--128 (2007; Zbl 1171.11033)] given an Hida family of modular forms, weight \(2\) specialisations arise as sections of the cohomology of a modular curve of \(p\)-power level; the explicit description of Heegner points allows the author to \(p\)-adically interpolate in weight \(2\), obtaining the so called \textit{Big Heegner points}. This construction has been generalized by \textit{O. Fouquet} [Compos. Math. 149, No. 3, 356--416 (2013; Zbl 1286.11066)] and \textit{M. Longo} and \textit{S. Vigni} [Manuscr. Math. 135, No. 3--4, 273--328 (2011; Zbl 1320.11054)]. One of the main questions, solved by \textit{F. Castella} [J. Inst. Math. Jussieu 19, No. 6, 2127--2164 (2020; Zbl 1454.11108)] is the relation between higher weigh specialisations of Big Heegner points and classical Heegner cycles. The author proposes a new approach to the construction of classes in the Galois representation of Hida families over totally real number fields \(F\). The main novelty, already in the case \(F=\mathbb{Q}\), is the systematic use of powerful representation-theoretic techniques which both simplify the original construction of Howard and make results such as explicit reciprocity laws more approachable; not only this, but the abstract viewpoint makes it possible to generalise similar construction to othe cases (essentially, whenever one has a sufficiently well developed eigenvariety machine). The long introduction may serve as a good perspective on the several results contained in this important paper. In the introduction, after proving a general \(p\)-adic Gross-Zagier formula (Theorem B, whose consequences are recalled in Theorem A) the author constructs \(p\)-adic Heegner cycles attached to a Hida family by using eigenvariety techniques, and proves the interpolation property in Theorem C, which, from the new perspective of the paper, is very natural. A form of the explicitly reciprocity law is then given in Theorem D, and the arithmetic applications are given in Theorem E. Similar results have been obtained (with a similar approach) by \textit{D. Jetchev} et al. [Proc. Lond. Math. Soc. (3) 122, No. 1, 124--152 (2021; Zbl 1477.11108)], \textit{D. Loeffler} [J. Théor. Nombres Bordx. 33, No. 3, Part 2, 1021--1043 (2022; Zbl 1497.11130)] and \textit{K. Büyükboduk} and \textit{A. Lei} [J. Lond. Math. Soc., II. Ser. 104, No. 4, 1682--1716 (2021; Zbl 1500.11079)].
scientific article; zbMATH DE number 7601286
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