Greenberg's L-invariant of the symmetric sixth power of an ordinary modular form

L-invariants arose in the work of Mazur, Tate, and Teitelbaum on a p-adic analogue of the Birch and Swinnerton-Dyer conjecture. They found that, for certain elliptic curves, the associated p-adic L-function vanishes even when the usual L-function does not. They then conjectured the existence of the so-called L-invariant of such an elliptic curve allowing to recuperate the interpolation property of the p-adic L-function, but for its derivative instead. Greenberg and Stevens proved this conjecture by relating the L-invariant to a derivative of a Frobenius eigenvalue varying in a Hida family. Greenberg subsequently gave a candidate for L-invariants of p-adic L-functions of higher rank motives in terms of Galois cohomology.

Using a functoriality result of Ramakrishnan and Shahidi, we obtain a formula for Greenberg's L-invariant of the symmetric sixth power of an ordinary non-CM modular form of even weight greater than two. To do so, we construct a global Galois cohomology class using a method reminiscent of Ribet's. The formula relates the L-invariant to derivatives of p-adic analytic functions interpolating Frobenius eigenvalues varying in the Hida family on GSp(4) attached to the symmetric cube of the modular form.