The L-invariant of the symmetric square of an ordinary elliptic curve

In the mid '80s, Mazur, Tate, and Teitelbaum conjectured the existence of a local invariant—the L-invariant—attached to elliptic curves that are split multiplicative at p. They had found that for such elliptic curves the associated p-adic L-function necessarily vanished at s = 1. The L-invariant was then conjectured to relate the value of the derivative of the p-adic L-function at s = 1 with the value of the usual archimedean L-function. Greenberg and Stevens proved this conjecture using Hida theory to add a variable to the p-adic L-function. A critical part of their proof involves reformulating the L-invariant and relating it to the Hida deformation. Soon afterwards, Greenberg gave a definition of the L-invariant for very general p-ordinary Galois representations in terms of Galois cohomology.

As a first step in an attempt to generalize the Greenberg–Stevens result to higher rank groups, we relate Greenberg's L-invariant of the symmetric square of an ordinary elliptic curve to the Hida deformation attached to the elliptic curve. What complicates matters is that the L-invariant of a good ordinary elliptic curve is global and thus requires the construction of a global Galois cohomology class with certain prescribed local behaviour. We will present our construction, which is inspired by that of Ribet's, and also discuss a strategy for dealing with further cases.