This delightful book by Henri Darmon provides an updated summary of many of the recent developments in the arithmetic of elliptic curves and introduces the reader to the author’s new striking contributions to the subject.

The ultimate goal of the material presented in this book is to understand the structure of the group of rational points of elliptic curves E over number fields F, which is known to be abelian and finitely-generated by a classical theorem of Mordell-Weil.

The torsion subgroup of E(F) is already fairly well understood. Thanks to the remarkable results of Mazur, Merel and Parent, there exist methods for explicitly computing it and its order is uniformly bounded in terms of the degree of F.

On the other hand, the rank of E(F) has proved to be a much less feasible invariant of the curve; its behaviour and the conjectural relationship that it should bear with subtle arithmetic and analytic objects attached to E still remain an intriguing open problem. Important progress has been made in this direction in the last few decades and it is the aim of this book to report these developments.

The Shimura-Taniyama-Weil Conjecture, now a theorem due to the fundamental breakthrough of Wiles et al., asserts that every elliptic curve E over the field Q of rational numbers is modular. This amounts to saying that the L-function L(E, Q, s) of E equals the L-function L(f, s) of a normalized newform f of weight 2 and level N, the conductor of E. In turn, this allows one to prove that L(E, Q, s), which converges for Re(s) > 3/2, admits an analytic continuation and satisfies a functional equation of the form L*(E, Q, s) = sign(E, Q)·L*(E, Q, 2-s), where L*(E, Q, s) = (2π)-s Γ(s) Ns/2 L(E, Q, s) and sign(E, Q) = ±1. This material is quickly reviewed in chapters 1 and 2.

The analytic continuation and functional equation of L(E, F, s) is conjectured to hold for all elliptic curves E over any number field F. With a suitable notion of modularity, the ideas of Wiles lead to the proof of this fact when F is totally real and under certain restrictions on E.

The weak conjecture of Birch and Swinnerton-Dyer predicts that the rank of E(F) equals the order of vanishing of L(E, F, s) at s = 1. The strong version of the conjecture provides in addition a conjectural expression of the value of the leading term of L(E, F, s) at s = 1 in terms of the order of the (conjecturally finite) Tate-Shafarevic group, the regulator and the local Tamagawa numbers of E.

As explained in detail in Darmon’s book, the best theoretical evidence so far of the Birch and Swinnerton-Dyer Conjecture are the theorems of Gross-Zagier and Kolyvagin for elliptic curves E over Q, which prove the weak Birch and Swinnerton-Dyer Conjecture for E and the finiteness of the Tate-Shafarevic group provided that ords=1L(E, Q, s) ≤ 1. As discussed in chapter 7, Zhang has generalised these results to elliptic curves over totally real number fields.

The main ingredient of the proof of the theorems of Gross-Zagier and Kolyvagin is the existence of a supply of rational points on E over a tower of abelian extensions of an imaginary quadratic extension K of Q, a so-called non-trivial Heegner system attached to (E, K).

In order to construct such a Heegner system on E, one first manufactures it by means of the theory of complex multiplication on a suitable modular curve or a Shimura curve X attached to a quaternion algebra, and then projects it onto E through a modular parametrisation Φ: X→E. As explained in this monograph, such a parametrisation may be complex (as classically) or p-adic rigid analytic (within the theory of Mumford curves, thanks to the theorem of Cerednik-Drinfeld).

This construction is reported in its various flavours through chapters 3 to 6. Chapter 7 is devoted to the extension of this circle of ideas to elliptic curves over totally real number fields F and Heegner systems attached to purely imaginary quadratic extensions K of F.

It is a crucial observation that there exist Heegner systems attached to an imaginary quadratic field K if and only if sign(E, K) = -1. For an arbitrary quadratic extension K/F of number fields, define S(E, K) to be the set of archimedean places of K and finite places of K at which E acquires split multiplicative reduction. When F = Q and E has semistable reduction, one has sign(E, K) = (-1)#S(E, K) and the same is expected to hold for any K/F.

A refinement of the Birch and Swinnerton-Dyer conjecture leads to the prediction that there exists a non-trivial Heegner system for a semistable elliptic curve E over F and a quadratic extension K/F if and only if #S(E, K) is odd. Although this conjecture does not mention modularity at all, any attack to this problem should probably involve the connection between E and automorphic forms of some sort. Note for instance that no classical result sheds light on this question when F is not totally real or K admits some real archimedean place.

Chapter 10, which is somewhat independent of the rest of the book, describes the proof of Kolyvagin’s theorem, which shows the finiteness of the Tate-Shafarevic group whenever the elliptic curve is equipped with a suitable non-trivial Heegner system.

Chapters 8 and 9 contain the main new contributions of the author, which are alluded to at the end of chapters 3 and 7. For a modular elliptic curve E over a totally real number field F, these concern the (conjectural) construction of Heegner systems on E over abelian extensions of certain quadratic extensions K of F. More precisely, chapter 8 attacks the problem when K is an almost totally real (ATR) extension of F, i.e., a quadratic extension of F which is complex at a single archimedean place of F and real at the remaining places. In this case, the conjectural Heegner system on E is constructed by means of a suitable substitute of the classical uniformization of E by Poincaré’s upper half plane. Chapter 9 deals with the case of a real quadratic extension K of Q, provided sign(E, K) = -1. Under this assumption, there exists at least one prime factor p of the conductor N of E which does not split in K. Darmon's construction of a conjectural Heegner system attached to (E, K) is based on the same principles of the previous chapter, where now rigid analytic uniformization at p plays the role that complex uniformization did before.

This rather short monograph is written in a style that allows the reader to understand the guidelines of an exciting and involved subject, gathering in a unified context a good number of deep results that may seem heterogeneous to a non-specialist. This is of course at the cost of omitting many details, which are often relegated to the references or the list of exercises.