On the Iwasawa \(\mu\)-invariant and \(\lambda\)-invariant associated to tensor products of newforms (Q6543097)

Fix an odd prime number \(p\), and let \(\overline{\rho}_1,\dots,\overline{\rho}_t\) be a collection of two-dimensional ordinary Galois representations defined over a finite field \(\mathbb F_{p^e}\). Suppose that we are given newforms \(f_1,\dots,f_t\) whose \(p\)-adic representations \(\rho_{f_1}: G_{\mathbb Q} \to \text{GL}_2(\mathcal K)\), \(\cdots\), \(\rho_{f_t}: G_{\mathbb Q} \to \text{GL}_2(\mathcal K)\), with \(\mathcal O_{\mathcal K}/\pi_{\mathcal K} \cong \mathbb F_{p^e}\), satisfy \(\rho_{f_1} \otimes \cdots \otimes \rho_{f_t} \mod \pi_{\mathcal K} \cong \overline{\rho}_1 \otimes \cdots \otimes \overline{\rho}_t\) (with some extra technical hypotheses H1 - H4 on page 457). Here \(\mathcal K\) is a finite extension of \(\mathbb Q_p\) containing all the Fourier coefficients of the \(f_i\)'s, and \(\pi_{\mathcal K}\) is a chosen uniformizer in \(\mathcal K\). The author determines the cyclotomic \(\lambda\)-invariant for the Selmer group attached to the product \(f_1 \otimes \cdots \otimes f_t\) under the assumption that the \(\mu\)-invariant is zero. The main results are formulated in Theorems 2.1, 2.2, and 2.3.\N\NIf \(t=2\) this allows to deduce the Iwasawa Main Conjecture for \(f_1 \otimes f_2\) if it is already known for a congruent pair \(f'_1 \otimes f'_2\) much as \textit{R. Greenberg} and \textit{V. Vatsal} did for \(t=1\) [Invent. Math. 142, No. 1, 17--63 (2000; Zbl 1032.11046)].\N\NProofs of the main results combine a whole array of deep results and constructions concerning Galois representations, Selmer groups, Hecke algebras, and \(p\)-adic \(L\)-functions.