Variation of the algebraic λ-invariant over a solvable extension. (21st May 2021) - DELBOURGO, DANIEL
Abstract: Fix an odd prime p . Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(μ_{p^{∞}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{×}\ $ where g > 0 and $0 \lt n'≤ n$ . Given a modular Galois representation $\overline{ρ}:G_{\mathbb{Q}}→ \text{GL}_2(\mathbb{F})$ which is p -ordinary and also p -distinguished, we shall write $\mathcal{H}(\overline{ρ})$ for the associated Hida family. Using Greenberg's notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} λ^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{Λ}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{ρ})$, under the assumption $μ^{\text{alg}}_{K(μ_p)}(f_0)=0$ for at least one form f 0 . We can then easily deduce that $λ^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{ρ})$, generalising a theorem of Emerton, Pollack and Weston for $λ^{\text{alg}}_{\mathbb{Q}(μ_{p})}(f)$ . For example, if $\mathcal{D}_{∞}=\bigcup_{n≥ 1}\mathcal{D}_n$ has the structure of a p -adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{∞}/K$ is a g -fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{∞}/K(μ_p)\big)$ has dimension ≤ 3 and is a pro- p -group.