I would like to ask if there exist pedagogical expositions of the MordellWeil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to read a proof of MordellWeil?

1$\begingroup$ I cleaned up the wikipedia link. $\endgroup$ – David Roberts Aug 8 '11 at 6:41

2$\begingroup$ For elliptic curves over $\mathbb{Q}$, the proof is quite elementary, especially if you assume that the points of order 2 are rational. For elliptic curves over a number field, you need to know the finiteness of the class number and the finite generation of the group of units (basic facts in algebraic number theory). For abelian varieties, you need to know rather a lot of algebraic geometry. You should be able to find what you want in online lecture notes. $\endgroup$ – anon Aug 8 '11 at 17:01

9$\begingroup$ There are already eight (perfectly fine) references in the answers. I wanted to comment that, apart from different emphases on various parts or a choice of heavy machinery vs computation, these are all the same proof. Namely, prove weak MW by finiteness of division fields, construct heights and do the descent argument. I wonder if there is a really different proof of MW. $\endgroup$ – Felipe Voloch Aug 8 '11 at 19:50
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 MordellWeil, NagellLutz Theorem, elliptic curves over finite fields, etc.
For more advanced treatment of MordellWeil, I suggest the following textbook:
J. Silverman "The arithmetic of elliptic curves" (Chapter 8 is about MordellWeil).
For the case of elliptic curves, there is Mordell's proof, discussed in his book Diophantine Equations (pp. 138148). I could hardly imagine less prerequisites than this.

2$\begingroup$ After reading this proof, I never understood why other proofs looked so complicated. $\endgroup$ – zeb Oct 29 '12 at 21:42

4$\begingroup$ This proof only considers elliptic curves over $\bf Q$ whose points of order 2 are $\bf Q$rational. This gives the following simplifications: * the algebraic geometry of Abelian varieties is reduced as its minimum; * no algebraic number theory is required; * the height machine is elementary; * the Galois cohomology construction can be treated by hand. That's why the general proof is more complicated. $\endgroup$ – ACL Nov 12 '12 at 23:23
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 MordellWeil 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).

3$\begingroup$ A belated comment: I am currently teaching a course on elliptic curves, primarily out of Silverman's first text (which is, of course, wonderful). I used Cassels' LEC as a supplementary text, and I have to say that the proof of MordellWeil (for elliptic curves over number fields) is where LEC really shines. He makes a beeline to MordellWeil and gives a simple, but not overly computational proof, in an impressively short span of pages. $\endgroup$ – Pete L. Clark Oct 29 '12 at 18:51

3$\begingroup$ Especially, there is a part of the proof of MordellWeil which is traditionally proved using aspects of the reduction theory of elliptic curves over local fields. (There are other ways, e.g. ChevalleyWeil, but I decided to bypass them for various reasons.) Silverman devotes an entire chapter to elliptic curves over local fields and another entire chapter to formal groups, to prove the key fact that the kernel of reduction contains no torsion of order prime to the residue characteristic. Cassels gives a simply beautiful proof of this that takes about two pages. $\endgroup$ – Pete L. Clark Oct 29 '12 at 18:54

$\begingroup$ @Pete L. Clark : thank you for pointing to Cassels LEC. One minor issue is that the proof of the key fact is performed only over $\mathbb Q_p$. I found the same proof worked out for a general local field in : Henri Cohen Number Theory Volume I: Tools and Diophantine Equations GTM 239 §7.3.67. I even found the exposition somewhat better. $\endgroup$ – Niels Jan 5 '17 at 13:35

$\begingroup$ @Niels: I do think it's minor. In fact, here are my notes for the course: math.uga.edu/~pete/EllipticCurves.pdf. In Section 9.6 I give Cassels's argument, adapted to any padic field (and even slightly more generally). I think it's a nice argument. That said, I am certainly a fan of Cohen's exposition as well, and it's nice to have a more formal reference for this argument. $\endgroup$ – Pete L. Clark Jan 6 '17 at 4:54
There is a very elementary and selfcontained (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.

11$\begingroup$ On the other hand one might also object that it's misleading to use "intrinsic facts from algebraic geometry" without explaining how these naturally generalize explicit techniques going back to Fermat. Those techniques aren't even that unnatural or obscure: the only tricky point is that on $y^2 = x (x^2 + ax + b)$ the $x$coordinate gives a homomorphism from the elliptic curve group to $k^*/(k^*)^2$, and that was probably observed experimentally before looking for a proof by polynomial identities  which still happened long before it could be interpreted in terms of Galois cohomology. $\endgroup$ – Noam D. Elkies Aug 7 '11 at 22:58

$\begingroup$ I completely agree! (Which is why I said "might" and "can"...). And Ireland and Rosen give many references; a student following them gets a very good motivated introduction to Galois cohomology... $\endgroup$ – Denis Chaperon de Lauzières Aug 8 '11 at 5:10
Manin's proof of MordellWeil 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.

$\begingroup$ An extensive list of corrections, by the venerable MOer BCnrd, to errors introduced in the TeXed version can be found here math.tifr.res.in/~publ/abelianvarietieserrata.pdf $\endgroup$ – Chandan Singh Dalawat Aug 8 '11 at 2:55

$\begingroup$ Professor, just want to mention a small technicality I read in the proof by Manin. He assumed $h(x,x)$ is discrete on $\Gamma$ because of CauchySchwartz inequality and $\Gamma/n\Gamma$ is finitely generated. I do not really understand his reasoning. I think the descent argument itself does not really need the assumption $h(x,x)$ is discrete. $\endgroup$ – Bombyx mori Apr 5 '17 at 6:31
Here is a short proof of the weak MordellWeil 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 HindrySilverman, or in [Brian Conrad, http://math.stanford.edu/~conrad/papers/Kktrace.pdf ], section 9 (Conrad even proves a more general theorem, the LangNéron 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 TateShafarevich 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 HochschildSerre 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: $H^1(X',\mathbf{G}_m) = \mathrm{Pic}(X') = \mathrm{Cl}(X')$).
Actually, the wikipedia article you cite cites Joe Silverman's book, which contains such a "pedagogical" exposition. The book is not entirely selfcontained, but I am sure the preface explains the prerequisites.

$\begingroup$ The standard proof of the weak Mordell theorem in Silverman unnecessarily translates the "putative finiteness of $E(L)/nE(L)$ into a statement about certain field extensions of $L$". There are simpler proofs that directly relate the finiteness to the finiteness of the class number of $L$ and the finite generation of its units. $\endgroup$ – anon Aug 8 '11 at 0:34

2$\begingroup$ I did not say Silverman had the BEST possible proof (indeed, I am sure opinions vary on what the best proof is),but it IS pedadgogical, which is all the OP asked for, and the reference was staring him in the face (since he was quoting the wiki article). $\endgroup$ – Igor Rivin Aug 8 '11 at 2:22
I think one should also mention
Jean Pierre Serre
Lectures on the MordellWeil Theorem
Aspects of Mathematics

$\begingroup$ This is one of the best books available on the subject, but it is certainly not the easiest. (Of course it is still "pedagogical", but it seems that the OP is looking for something with minimal prerequisites.) $\endgroup$ – Pete L. Clark Aug 7 '11 at 21:31
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 MordellWeil 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 MordellWeil 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).