Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms

Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

On Greenberg’s L-invariant of the symmetric sixth power of an ordinary cusp form

We derive a formula for Greenberg’s L-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight ≥4, under some technical assumptions. This requires a ‘sufficiently rich’ Galois deformation of the symmetric cube, which we obtain from the symmetric cube lift to GSp(4)/Q of Ramakrishnan–Shahidi and the Hida theory of this group developed by Tilouine–Urban. The L-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg’s L-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.

Twisted Gross–Zagier Theorems

The theorems of Gross–Zagier and Zhang relate the Néron–Tate heights of complex multiplication points on the modular curve X0(N) (and on Shimura curve analogues) with the central derivatives of automorphic L-function. We extend these results to include certain CM points on modular curves of the form X (Ⲅ0(M ) ∩ Ⲅ1(S)) (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article “Central derivatives of L -functions in Hida families”,Math. Ann. 399(2007), 803–818.

ON TRACIAL DISTRIBUTIONS — A CONTRIBUTION TO THE HIDA THEORY

Tracial distributions are defined as the elements of the dual of a countably Hilbertian space and on which a projective limit topology is defined. The connection of these distributions with Hida's generalized functions is discussed.