Iwasawa theory of automorphic representations of \(\mathrm{GL}_{2n}\) at non-ordinary primes (Q2105826)

Let \(p\) be a fixed odd prime and \(f = \sum_{m \geq 1} a_m q^m\) a cuspidal elliptic modular form. When \(f\) has good ordinary reduction at \(p\), \textit{B. Mazur} and \textit{P. Swinnerton-Dyer} [Invent. Math. 25, 1--61 (1974; Zbl 0281.14016)], as well as \textit{Yu. I. Manin} [Mat. Sb., Nov. Ser. 92(134), 378--401 (1973; Zbl 0293.14007)], constructed a \(p\)-adic \(L\)-function attached to \(f\) which is a bounded measure on \(\Gamma := \mathrm{Gal}(\mathbb Q(\mu_{p^{\infty}})/\mathbb Q)\), interpolating complex \(L\)-values of \(f\) twisted by Dirichlet characters on \(\Gamma\). In the non-ordinary case, the resulting \(p\)-adic \(L\)-functions are not necessarily bounded (but are \(h\)-admissible in the sense of \textit{Y. Amice} and \textit{J. Vélu} [Astérisque 24--25, 119--131 (1975; Zbl 0332.14010)] and \textit{M. M. Vishik} [Mat. Sb., Nov. Ser. 99(141), 248--260 (1976; Zbl 0358.14014)]). \textit{R. Pollack} [Duke Math. J. 118, No. 3, 523--558 (2003; Zbl 1074.11061)] showed that when \(a_p = 0\), there is a very elegant way to decompose the \(p\)-adic \(L\)-functions attached to \(f\), using the so-called plus and minus logarithms, into bounded measures by exploiting the symmetry between the two roots of the Hecke polynomial of \(f\) at \(p\). These bounded measures were utilized by \textit{S.-i. Kobayashi} to formulate the so-called plus and minus Iwasawa main conjectures in [Invent. Math. 152, No. 1, 1--36 (2003; Zbl 1047.11105)] when \(f\) corresponds to an elliptic curve defined over \(\mathbb Q\). Now let \(\Pi\) be any cuspidal automorphic representation of \(\text{GL}_{2n}(\mathbb A_{\mathbb Q})\), and let \(p\) be an odd prime at which \(\Pi\) is unramified. In a recent work [``On $p$-adic $L$-functions for GL$_{2n}$ in finite slope Shalika families'', Preprint, \url{arXiv:2103.10907}], \textit{D. Barrera} et al. constructed (possibly unbounded) \(p\)-adic \(L\)-functions interpolating complex \(L\)-values of \(\Pi\) in the non-ordinary case. Under certain assumptions, the authors construct two bounded \(p\)-adic \(L\)-functions for \(\Pi\) (see Theorem B), thus extending an earlier work of Rockwood [Ann. Math. Que., to appear] by relaxing the Pollack condition. In Conjecture 5.5, the authors formulate the Iwasawa main conjectures connecting the Pontryagin duals of the signed Selmer groups with the bounded \(p\)-adic \(L\)-functions given by Theorem B (a generalization of Kobayashi's main conjectures).