On obstructions to the Euler system method for Rankin-Selberg convolutions

Abstract To apply the Euler system method to a p -adic Galois representation V , one needs the existence of some $$\sigma \in G_{\mathbb {Q}(\mu _{p^{\infty }})}$$ σ ∈ G Q ( μ p ∞ ) such that $$V/(\sigma -1)V$$ V / ( σ - 1 ) V is free of rank one over the coefficient ring. Such a $$\sigma $$ σ is called an Euler-suitable element for V . Given a non-CM classical newform f of weight $$k \ge 2$$ k ≥ 2 and a weight one newform g with characters $$\chi ,\psi $$ χ , ψ , a prime ideal $$\mathfrak {p}$$ p of residue characteristic p of a suitably large number field, we consider the situation where $$V=V_{f,g,\mathfrak {p}}$$ V = V f , g , p is the tensor product of the $$\mathfrak {p}$$ p -adic representations attached to f and g . D. Loeffler asked the following question: is it true that, if $$\chi \psi \ne 1$$ χ ψ ≠ 1 , then V has an Euler-suitable element for all but finitely many p ? He also gave an affirmative answer when f , g had coprime conductors. We give several weaker sufficient conditions to answer this question in the affirmative, and we can as an application remove some of the technical assumptions required in the statement of the Birch- and Swinnerton-Dyer conjecture proved by Kings, Loeffler and Zerbes. We also show that the general answer to the question is negative, by constructing families of counter-examples, and giving additional counter-examples that do not fit in this family.