Variation of the analytic λ ‐invariant over a solvable extension

Fix an odd prime p , and suppose thatD ∞ = ⋃ n D nis a solvable p ‐adic Lie extension of the rationals, such that Gal ( D n / Q ) ≅ ( Z / p n ′ Z ) g ⋊ ( Z / p n Z ) ×for some g > 0 andn ′ ⩽ n . In particular, this situation covers the well‐known cases where (i)D ∞ = Q ( μ p ∞ , Δ 1 1 / p ∞, ⋯ , Δ g 1 / p ∞)is a g ‐fold false‐Tate tower, or (ii) dim ( G ∞ ) ⩽ 3 withG ∞ = Gal ( D ∞ / Q )a non‐commutative Lie group. Let H ( ρ ¯ ) be the Hida family associated to a modular Galois representationρ ¯ : G Q → GL 2 ( Fp e) . We derive an exact formula for the analytic λ ‐invariantλ D n an ( f ) = the number of zeroes ofL p f / D n , Tat every f ∈ H ( ρ ¯ ) , and over each finite layer D n in the p ‐adic Lie extension, under the assumptionμ Q ( μ p ) an ( f 0 ) = 0 for at least one form f 0 in the family. We can then easily deduce thatλ D n an ( f )is constant along branches of H ( ρ ¯ ) , thereby extending a theorem of Emerton, Pollack and Weston forλ Q ( μ p ) an ( f )to a non‐commutative Iwasawa theory setting.