| Let f ∈ Sk(Gamma 0(N), psi) be a p-ordinary newform and r f the associated Galois representation. We find the special value Lp(1, r f ⊗ rf&d15; ). We define the analytic L -invariant of a "motivic" Galois representation, and show how this special value relates to work of Greenberg and Hida on finding L p(Ad( r f)). In particular, we reduce finding this value to showing an equality of p-adic L-functions similar to a well-known relation of archimedean L-functions.;Given a Hecke character psi for an imaginary quadratic field K, let f be the theta series corresponding to psi. We show that one has an equality Lp(s , Ad(f)) = Lp( s, aK/Q ) · Lp(s, psi -) corresponding to the well-known decomposition INDQK (psi) = aK/Q ⊗ Ind(psi-), where aK/Q , is the Dirichlet character corresponding to the extension K/ Q and psi- is the "anticyclotomic part" of the character psi. Using our computation above and theorems of Gross [Gro80] and Hida [Hid07], this leads to a formula for the L -invariant of the representation Ad(f), which is exactly the value conjectured by Hida in [Hid04]. This also gives a new proof of the Ferrero-Greenberg theorem in the case of a quadratic Dirichlet character. |
