Iwasawa theory for symmetric squares of non-\(p\)-ordinary eigenforms (Q2228546)

Let \(p \geq 7\) be a prime. In joint work with \textit{D. Loeffler} [Algebra Number Theory 13, No. 4, 901--941 (2019; Zbl 1443.11226)] the authors investigated the Iwasawa theory of the Rankin-Selberg convolution of two non-\(p\)-ordinary modular forms \(f\) and \(g\). In this paper, they focus on the case \(f=g\) thereby extending work of \textit{D. Loeffler} and \textit{S. L. Zerbes} [J. Reine Angew. Math. 752, 179--210 (2019; Zbl 1430.11149)] in the ordinary case. We now explain their results in more detail, but we suppress most of the technical hypotheses. Let \(f\) be a normalised cuspidal eigen-newform of weight \(k+2\), level \(N\) and nebentype \(\varepsilon\). Let \(\pm \alpha\) be the roots of the Hecke polynomial of \(f\) at \(p\) and let \(L\) be a number field containing the Hecke field of \(f\) and \(\alpha^2\). Let \(E\) be the completion of \(L\) at a prime \(\mathfrak{p}\) above \(p\) with ring of integers \(\mathcal{O}\). Let \(W_f^{\ast}\) be the dual of the associated (that is Deligne's) \(E\)-linear representation \(W_f\) of \(G_{\mathbb{Q}}\). For each pair \(\lambda, \mu \in \{\pm \alpha\}\) their is a collection of Beilinson-Falch classes \(BF_m^{\lambda, \mu}\) (see [\textit{D. Loeffler} and \textit{S. L. Zerbes}, Res. Math. Sci. 3, Paper No. 29, 53 p. (2016; Zbl 1414.11070)]). Taking a twist of these classes by an even Dirichlet character \(\chi\) gives a collection of compatible, but non-integral cohomology classes with values in \(\mathrm{Sym}^2 W_f^{\ast}(1+\chi)\). However, in order to use the machinery of Euler systems one needs integral systems and this is one of the main tasks tackled in this article. Let \(R_f\) be a Galois-stable \(\mathcal{O}\)-lattice in \(W_f\) an put \(T := \mathrm{Sym}^2 R_f^{\ast}(1+\chi)\). Building on their earlier work with Loeffler, the authors define three signed classes in the Iwasawa cohomology of \(T\) (a forth class is identically zero). The running assumptions imply that there is a \(G_{\mathbb{Q}_p}\)-equivariant decomposition \[ T = R_{1,\chi}^{\ast} \oplus R_{2,\chi}^{\ast}. \] Following \textit{A. Lei} [Glasg. Math. J. 54, No. 2, 241--259 (2012; Zbl 1300.11113)] this allows the definition of three signed Coleman maps \[ H^1_{\mathrm{Iw}}(\mathbb{Q}_p(\mu_{p^{\infty}}), T) \rightarrow \mathcal{O}[[\Gamma]], \] where \(\Gamma := \mathrm{Gal}(\mathbb{Q}(\mu_{p^{\infty}}) / \mathbb{Q})\). The images of the signed classes under the signed Coleman maps are then doubly signed Beilinson-Flach \(p\)-adic \(L\)-functions. The kernels of these maps are used to define certain local Selmer conditions at \(p\). So this leads to the definition of doubly signed Selmer groups and the formulation of Iwasawa main conjectures in the usual way. Some evidence for these conjectures is also provided. Let \(j\) be an integer and let \(e_{\omega^j}\) be the idempotent associated to the \(j\)-th power of the Teichmüller character \(\omega\). Then for even \(k+2 \leq j \leq 2k+2\) there is a choice of double sign such that the \(e_{\omega^j}\)-part of the corresponding \(p\)-adic \(L\)-function \(\mathcal{L}\) does not vanish. Moreover, the corresponding Selmer group is co-torsion and the characteristic ideal of its dual divides \((e_{\omega^j} \mathcal{L})\) (in \(\mathcal{O}[[T]] \otimes \mathbb{Q}_p\)). \par For this one has to show that at least one of the Euler systems is non-trivial. Here the authors refer to (at that time) unpublished work of \textit{A. Arlandini}. The required result is now available as preprint with \textit{D. Loeffler} [``On the factorisation of the \(p\)-adic Rankin-Selberg \(L\)-function in the supersingular case'', Preprint, \url{arXiv:2103.16380}]. Finally, the authors prove a partial result towards an `analytic' main conjecture in the spirit of \textit{J. Pottharst} [Algebra Number Theory 7, No. 7, 1571--1612 (2013; Zbl 1370.11123)] and \textit{D. Benois} [Lond. Math. Soc. Lect. Note Ser. 420, 36--88 (2015; Zbl 1388.11078)].