On the image of \(\Lambda\)-adic Galois representations
Let \(f\) be a cuspidal Hecke-eigenform without complex multiplication, \(E\) the number field generated by its Hecke-eigenvalues, \(\mathcal O\) the ring of integers in \(E\). Following \textit{F.\ Momose} [J.\ Fac.\ Sci., Univ.\ Tokyo, Sect.\ I A 28, 89-109 (1981; Zbl 0482.10023)], the author attaches to \(f\) a certain subfield \(\tilde E\) of \(E\) and a subgroup \(H\) of finite index inside the absolute Galois group \(G\) of the rationals. The author shows that there exists a set \(\Sigma\) of prime numbers which has density 1 and only contains primes \(l\) where \(f\) is ordinary such that for its family of \(\Lambda\)-adic forms (where \(\Lambda = \mathbb{Z}_l T\)) in the sense of \textit{H.\ Hida} [Invent. Math. 85, 545-613 (1986; Zbl 0612.10021)] the following holds: If \(f\) is \(\ell\)-ordinary for a prime \(\ell\) of \(E\) above \(l\in\Sigma\), then the \(\Lambda\)-adic ordinary Hecke algebra above \(\ell\) is isomorphic to \({\mathcal O}_\ellT\) and the image of \(H\) under the \(\Lambda\)-adic Galois-representation attached to \(f\) contains SL\(_2(\tilde{\mathcal O}_{\tilde\ell}T)\), where \(\tilde\ell = \ell\cap \tilde {\mathcal O}.\) This interesting result is additionally spiced up by a specialization result: if a \(\Lambda\)-adic modular form does not admit complex multiplication, then neither do its ``classical''specializations of weight at least 2.
scientific article; zbMATH DE number 1794802
scientific article