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Let $E:y^2=(x-\alpha )(x-\beta )(x-\gamma )$ with distinct algebraic integers in a number field $K$. For a rational point $P$, its halving field is unramified outside the finite set $S$ of primes dividing $2(\alpha -\beta )(\alpha -\gamma )(\beta -\gamma )$. At $v\notin S$, the elliptic-curve discriminant is a unit, so there is good reduction. A half of the reduction of $P$ is defined over a finite extension of the residue field. Pass to the corresponding unramified extension of the completion and lift this half using the Hensel lemma. The doubling error lies in the formal kernel of a minimal Weierstrass equation. Since two is a unit, prime-to-residue-characteristic multiplication on a formal group corrects this error uniquely. Every half differs by rational 2-torsion, so all halves are unramified at $v$. The halving cocycle with rational two-torsion has image of order at most four, making the halving field a Galois extension of degree at most four. The bounded-degree extensions with restricted ramification are finite in number; there are only finitely many homomorphisms from their finite Galois groups to $E[2]$. Thus $E(K)/2E(K)$ is finite, without assuming any form of the Mordell-Weil theorem.

The unramified Iwasawa module is finitely generated torsion over the Iwasawa algebra of a Zp-extension, for every Zp-extension of a number field. After a finite shift all ramified primes are totally ramified and their number $s$ is constant. Class field theory bounds finite-layer coinvariant modules by ${\mathrm{rank}}_{{\mathbb{Z}}_p}{Y}_{{\Gamma }_n}\le s-1$. The Compact Nakayama lemma proves finite generation, and a positive Iwasawa-module rank would force ranks at least $r{p}^n$, a contradiction. No Leopoldt conjecture is required.

For a number field $K$, a finite set $S$ and $m\ge 2$, let $K(S,m)$ consist of classes $[a]\in K^{\ast }/K^{\ast m}$ whose valuations outside $S$ are divisible by $m$. It fits into $0\to {\mathcal{O}}_{K,S}^{\times }/({\mathcal{O}}_{K,S}^{\times }{)}^m\to K(S,m)\to \mathrm{Cl}({\mathcal{O}}_{K,S})[m]\to 0$. The S-unit group is finitely generated and the ideal class group is finite, so $K(S,m)$ is finite. When $S$ contains primes dividing $m$, this controls unramified multiplicative Kummer classes outside $S$.

On a projective variety over a number field, a very ample line bundle gives a logarithmic projective height by an embedding. Tensor products correspond to sums by the Segre embedding, and a sufficiently ample twist expresses any line bundle as a difference of very ample ones. This defines its height modulo bounded functions. It is additive and functorial under morphism pullback. An ample bundle has the Northcott theorem finiteness property for bounded height and bounded point degree; a trivial bundle need not. The product formula makes these absolute heights independent of scalar coordinates and of the number field used to compute them.

A Zp-extension is an infinite Galois extension of a number field whose Galois group is topologically isomorphic to the additive p-adic integers. There is a unique intermediate field of degree $p^n$ for every $n\ge 0$, corresponding to $p^n{\mathbb{Z}}_p$. The finite layers form a tower with cyclic successive degree $p$ extensions.