The Exceptional Zero Conjecture for Symmetric Powers of CM Modular Forms: the Ordinary Case

Abstract

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic Lfunctions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.In an appendix, we prove a functional equation for some of the p-adic L-functions we construct.

Derivative of symmetric square p-adic L-functions via pull-back formula

“…We point out that the analytic L-invariant for CM forms has already been studied in literature [DD97,Har12,HL13]. Note also that our choice of periods is not optimal [Ros13b, §6].…”

Iwasawa Theory 2012

“…3. Symmetric powers of CM-modular forms (Harron 2013;Harron and Lei to appear). The associated p-adic representation V is either crystalline or potentially crystalline at p and we do need the theory of .…”

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

“…(b) The L-invariants of all symmetric powers in the p-ordinary CM case have been treated in [Har11].…”

“…Another exception comes again from Greenberg's original article [Gre94], where he computed his L-invariant when E has good ordinary reduction at p and has complex multiplication. In this case, the symmetric powers are reducible and the value of the L-invariant comes down to the result of Ferrero-Greenberg [FG78]; see the author's article [Har11] for the details in the more general case of a CM modular form. The general difficulty in the crystalline case is that Greenberg's L-invariant is then a global invariant and its computation requires the construction of a global Galois cohomology class.…”