Proofs of Mordell-Weil theorem

Question

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?

Answer 1

J. Silverman and J. Tate "The rational points on elliptic curves" is a wonderful introduction to elliptic curves over rational numbers. It covers topics such as Mordell-Weil, Nagell-Lutz Theorem, elliptic curves over finite fields, etc.

For more advanced treatment of Mordell-Weil, I suggest the following textbook:

J. Silverman "The arithmetic of elliptic curves" (Chapter 8 is about Mordell-Weil).

Answer 2

For the case of elliptic curves, there is Mordell's proof, discussed in his book Diophantine Equations (pp. 138-148). I could hardly imagine less prerequisites than this.

Answer 3

There must be a proof in Cassels' Lectures on elliptic curves (Cambridge University Press, Cambridge, 1991).

See also his masterly survey Diophantine equations with special reference to elliptic curves (J. London Math. Soc. 41 (1966) 193–291) and the historical essay Mordell's finite basis theorem revisited (Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 31–41).

Here is a quote from this last paper :

Weil's generalization of Mordell's theorem (and subsequent generalizations) was usually referred to as the Mordell-Weil Theorem. Mordell himself strongly disapproved of this usage and frequently insisted (in public and in private) that what he had proved should be called Mordell's Theorem and that everything else could, for his part, be called simply Weil's Theorem.

Addendum. Another excellent source is Knapp's Elliptic curves (Princeton University Press, Princeton, 1992) which contains a proof of Mordell's theorem (over $\mathbf Q$).

There is a very affordable book by Milne (Elliptic curves, BookSurge Publishers, Charleston, 2006) and a very motivating one by Koblitz (Introduction to elliptic curves and modular forms, Springer, New York, 1993). Tate's Haverford Lectures also served as the basis for Husemoller (Elliptic curves, Springer, New York, 2004).

Answer 4

There is a very elementary and self-contained (modulo a few things proved earlier in the book) proof in Chapter 19 of the book of Ireland and Rosen, "A classical introduction to modern number theory". One might object that it can be misleading to use explicit but obscure polynomial identities instead of more intrinsic facts from algebraic geometry, but the text has lots of good remarks and references to go beyond this elementary approach.

Answer 5

Manin's proof of Mordell-Weil theorem (for abelian varieties over number fields) has appeared as an appendix to Russian translation of First edition of Mumford's ``Abelian varieties". Eventually it was translated into English and published as an appendix to Second and Third editions of Mumford's book.

Answer 6

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: $H^1(X',\mathbf{G}_m) = \mathrm{Pic}(X') = \mathrm{Cl}(X')$).

Answer 7

Actually, the wikipedia article you cite cites Joe Silverman's book, which contains such a "pedagogical" exposition. The book is not entirely self-contained, but I am sure the preface explains the prerequisites.

Answer 8

I think one should also mention

Jean Pierre Serre

Lectures on the Mordell-Weil Theorem

Aspects of Mathematics

Answer 9

Already mentioned: Silverman and Tate's "Rational Points on Elliptic Curves" (undergraduate level) and Silverman's "The Arithmetic of Elliptic Curves" (graduate level).

Another text at the undergraduate level that covers Mordell's theorem (i.e., the Mordell-Weil theorem for elliptic curves over $\mathbb{Q}$) is Washington's "Elliptic Curves: Number Theory and Cryptography" (see Chapter 8).

If you are looking for a proof of the Mordell-Weil theorem in its utmost generality (i.e., for abelian varieties over number fields), I would suggest Hindry and Silverman's "Diophantine Geometry: An Introduction" (see Part C).