Abstract
For an ordinary prime p ⩾ 3 , we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q ( μ p ) whose Galois group is G ≅ Z p ⋊ Z p . For Selmer groups defined over the cyclotomic Z p -extension of Q ( μ p ) , we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.
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