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Greenberg's L-invariants of adjoint square Galois representations

Abstract

For a two-dimensional p-adic Galois representation V associated to a p-ordinary Hecke eigen cusp form f of weight k > 1, we identify the L-invariant (of R. Greenberg) of the (three dimensional) adjoint square Ad(V) of V with the derivative of the p-coefficient of the Lambda-adic lift of f. By this result, for a given p-adic analytic family of ordinary Hecke eigenforms, the L-invariant does not vanish for almost all members in the p-adic family (as expected).

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