The exceptional zero conjecture for symmetric powers of CM modular forms: the ordinary case

TL;DR

This paper proves the exceptional zero conjecture for symmetric powers of CM modular forms at ordinary primes, identifying trivial zeroes, computing L-invariants, and confirming their agreement with Greenberg's invariants.

Contribution

It establishes the conjecture for symmetric powers of CM forms at ordinary primes, providing explicit calculations of L-invariants and their correspondence with Greenberg's invariants.

Findings

Identified trivial zeroes of p-adic L-functions for symmetric powers.

Computed L-invariants explicitly for these forms.

Confirmed L-invariants match Greenberg's predictions.

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 L-functions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.