On the image of \(\Lambda\)-adic Galois representations. (English) Zbl 1020.11037

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 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 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 \({\mathcal O}_\ell[[T]]\) 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.


11F85 \(p\)-adic theory, local fields
11F80 Galois representations
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