Independence of the two CM criteria

Determine whether failures of the cyclotomic criterion based on the cyclotomic p-adic regulator and failures of the Coates–Liang–Sujatha critical Hecke L-value criterion are independent as the split prime varies.

Background

The paper compares two criteria for proving the vanishing of $(E/\mathbb{Q})[p^\infty]$: the cyclotomic criterion involving the normalized second Taylor coefficient and regulator, and the Coates–Liang–Sujatha criterion involving a critical Hecke L-value.

In the computed range, the two criteria fail at disjoint sets of primes. The paper notes that an independence heuristic would imply that, apart from finitely many split primes, at least one criterion should establish vanishing.