TITLE: Proofs of Mordell-Weil theorem
QUESTION [12 upvotes]: I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to read a proof of Mordell-Weil?

REPLY [7 votes]: Here is a short proof of the weak Mordell-Weil theorem for Abelian varieties over a number field using étale cohomology (easily adopted to finitely generated fields). The construction of the height paring can be found in Hindry-Silverman, or in [Brian Conrad, http://math.stanford.edu/~conrad/papers/Kktrace.pdf ], section 9 (Conrad even proves a more general theorem, the Lang-Néron theorem).
Let $K$ be a number field, $A/K$ be an Abelian variety and $S$ a finite set of places of $K$.  Let $X = \mathrm{Spec}\mathcal{O}_{K,S}$ and $\mathscr{A}/X$ the Néron model of $A/K$. By the Néron mapping property, it suffices to show that $\mathscr{A}(X)/n = A(K)/n$ is finite for some $S$ and $n > 1$.
By enlarging $S$ by the set of primes lying over $n$, one has a short exact Kummer sequence $0 \to \mathscr{A}[n] \to \mathscr{A} \to \mathscr{A} \to 0$, inducing in (étale) cohomology $0 \to \mathscr{A}(X)/n \hookrightarrow H^1(X,\mathscr{A}[n])$. So it suffices to show that $H^1(X,\mathscr{A}[n])$ is finite.  (This group is related to the Selmer group. The cokernel $H^1(X,\mathscr{A})[n]$ is related to the $n$-torsion of the Tate-Shafarevich group.)
There is a finite étale Galois covering $X'/X$ such that $\mathscr{A}[n] \times_X X' \cong (\mathbf{Z}/n)^{2g} \cong \mu_n^{2g}$. The Hochschild-Serre spectral sequence $$H^p(\mathrm{Gal}(X'/X), H^q(X',\mathscr{A}[n] \times_X X')) \Rightarrow H^{p+q}(X,\mathscr{A}[n])$$ induces $$0 \to H^1(\mathrm{Gal}(X'/X), H^0(X',\mathscr{A}[n] \times_X X')) \to H^1(X,\mathscr{A}[n]) \to H^0(\mathrm{Gal}(X'/X), H^1(X',\mathscr{A}[n] \times_X X')).$$ Since $\mathrm{Gal}(X'/X)$ and $H^0(X',\mathscr{A}[n] \times_X X')$ are finite, the left hand group is finite, so it suffices to show that $H^1(X',\mathscr{A}[n] \times_X X') \cong H^1(X',\mu_n^{2g})$ is finite.  But the short exact Kummer sequence $1 \to \mu_n \to \mathbf{G}_m \to \mathbf{G}_m \to 1$ induces $$1 \to \mathbf{G}_m(X')/n \to H^1(X',\mu_n) \to H^1(X',\mathbf{G}_m)[n] \to 0.$$  The left hand group is finite by the finite generation of the $S$-unit group, and the right hand group is finite by the finiteness of the $S$-class number (Hilbert 90: $H^1(X',\mathbf{G}_m) = \mathrm{Pic}(X') = \mathrm{Cl}(X')$).