Journal of the London Mathematical Society ( IF 1.121 ) Pub Date : 2021-01-12 , DOI: 10.1112/jlms.12428
Marti Roset; Victor Rotger; Vinayak Vatsal

The purpose of this article is proving the equality of two natural $L$‐invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic $L$‐invariant as a universal norm of a canonical $Z p$‐extension of $Q p$ associated to the representation. We relate it to a certain $2 × 2$ regulator of $p$‐adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic $L$‐invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear].

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