| We obtain formulae for Greenberg's L-invariant of symmetric square and symmetric sixth power motives attached to p-ordinary modular forms in the vein of theorem 3.18 of [GS93]. For the symmetric square of f, the formula obtained relates the L-invariant to the derivative of the p-adic analytic function interpolating the pth Fourier coefficient (equivalently, the unit root of Frobenius) in the Hida family attached to f. We present a different proof than Hida's, [Hi04], with slightly different assumptions. The symmetric sixth power of f requires a bigger p-adic family. We take advantage of a result of Ramakrishnan--Shahidi ([RS07]) on the symmetric cube lifting to GSp(4)/Q, Hida families on the latter ([TU99] and [Hi02]), as well as results of several authors on the Galois representations attached to automorphic representations of GSp(4) /Q, to compute the L-invariant of the symmetric sixth power of f in terms of the derivatives of the p-adic analytic functions interpolating the eigenvalues of Frobenius in a Hida family on GSp(4)/Q. We must however impose stricter conditions on f in this case. Here, Hida's work (e.g. [Hi07]) does not provide answers as specific as ours.;In both cases, the method consists in using the big Galois deformations and some multilinear algebra to construct global Galois cohomology classes in a fashion reminiscent of [Ri76]. The method employs explicit matrix computations. |
