\(\mathcal L\)-invariants of Tate curves
Let \(p\) be an odd prime and let \(F\) be a totally real number field. Let \(E\) be a Hilbert modular elliptic curve, without complex multiplication, defined over \(F\), and assume that \(E\) has multiplicative or good ordinary reduction at all primes dividing \(p\). Let \(\rho\) be the representation of \(\text{Gal}(\overline{F}/F)\) on the \(p\)-adic Tate module of \(E\). Under the assumption that a certain universal deformation ring is a power series ring, the author gives a formula for \({\mathcal L}(\text{Ind}_F^{\mathbb Q}\text{Ad}(\rho_E))\), where \({\mathcal L}\) is Greenberg's \({\mathcal L}\)-invariant, and he conjectures a formula for \({\mathcal L}(\text{Ind}_F^{\mathbb Q} (\text{Sym}^{\otimes n}(\rho_E)(-m))\), where \(n\geq 1\) and \(m\) is an integer such that the motive \(\text{Sym}^{\otimes n}(H^1(E))(-m)\) is critical at 1. The conjecture implies that the \({\mathcal L}\)-invariant should be essentially independent of \(n\). The present paper generalizes work of \textit{R. Greenberg} [Contemp. Math. 165, 149--174 (1994; Zbl 0838.11070)], of \textit{R. Greenberg} and \textit{G. Stevens} [Contemp. Math. 165, 183--211 (1994; Zbl 0846.11030)], and of \textit{B. Mazur, J. Tate} and \textit{J. Teitelbaum} [Invent. Math. 84, 1--48 (1986; Zbl 0699.14028)]. A different treatment of the present material, one that is more Galois cohomological and less automorphic, is given in subsequent works of the author [Publ. Res. Inst. Math. Sci. 45, No. 1, 1--24 (2009; Zbl 1173.11033)], [Int. Math. Res. Not. 2007, No. 23, Article ID rnm102, 49 p. (2007; Zbl 1183.11028)].
scientific article; zbMATH DE number 5635148