Karl Rubin Henri Darmon September 9 , 2007

1. Thaine’s “purely cyclotomic” method [Th88] for bounding the exponents of the ideal class groups of cyclotomic fields. The bounds that Thaine obtained were already known thanks to the proof of the Main Conjecture by Mazur andWiles, in which unramified abelian extensions of cyclotomic fields were constructed from reducible two-dimensional Galois representations occuring in the Jacobians of modular curves. Thaine’s method did not rely on modular curves, exploiting instead a norm-compatible system of units in abelian extensions of Q, the so-called cyclotomic or circular units which had already played a key role in Kummer’s investigations of the arithmetic of cyclotomic fields. Thaine’s ideas were transposed to great effect by the author of the monograph under review to the context of abelian extensions of imaginary quadratic fields, with the role of the circular units being played by the elliptic units of Siegel and Robert-Ramachandra. In [Ru87], the methods of Coates and Wiles were thus strengthened to give a proof of the finiteness of the Shafarevich-Tate group for complex multiplication elliptic curves with non-vanishing L-series at s = 1. This yielded the first examples of elliptic curves whose Shafarevich-Tate groups could be proved to be finite, a breakthrough which dramatically illustrated the power of Thaine’s point of view.