L-invariants of low symmetric powers of modular forms and Hida deformations

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