Analogues of Iwasawa's $μ=0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension

On the main conjecture of Iwasawa theory for certain non‐cyclotomic Zp‐extensions

“…As a consequence, we obtain that E(H ∞ ) and E(F ∞ ) modulo torsion are finitely generated abelian groups. This is discussed further in [3].…”

“…We remark also that the main conjecture for H ∞ /H is an important step to proving the main conjecture for F ∞ /F , which can be used to study the p-part of the Birch-Swinnerton-Dyer Conjecture for E/H. Furthermore, for p = 2, the construction of the p-adic L-function in Chapter 3 and the computation of the Iwasawa invariants in Chapter 5 of this paper are crucial for the proof in [3] that X(H ∞ ) is a finitely generated Z p -module (the case p > 2 was proven independently by Gillard [9,Theorem 3.4] and Schneps [17,Theorem IV]). This can be applied to prove the weak p-adic Leopoldt conjecture for certain non-abelian extensions.…”

“…As a corollary, Theorem 3.19 implies that X(H ∞ ) = 0 if and only if ϕ is a unit. It is known that (see [3,Section 5]), when p = 2, X(H ∞ ) = 0 for all primes q ≡ 7 mod 8 with q < 500 and q = 431. Thus Theorem 3.11 gives us information on the p-adic valuation of…”