##
**Euler systems. (Hermann Weyl lectures).**
*(English)*
Zbl 0977.11001

Annals of Mathematics Studies. 147. Princeton, NJ: Princeton University Press. xi, 227 p. (2000).

In 1986, F. Thaine [Ann. Math. (2) 128, 1-18 (1988; Zbl 0665.12003)] introduced a method for using cyclotomic units to bound ideal class groups of real abelian fields. Around the same time, V. Kolyvagin [Izv. Akad. Nauk SSSR, Ser. Mat. 52, 522-540 (1988; Zbl 0662.14017)] used Heegner points to construct cohomology classes, which were then used to bound Selmer groups of elliptic curves. Inspired by Thaine’s results and his own work, V. Kolyvagin [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017)] introduced the concept of an Euler system. Let \(K\) be a number field and let \(T\) be a finite-dimensional \(p\)-adic representation of \(\text{Gal}(\overline{K}/K)\). An Euler system is a collection of cohomology classes \(c_F\in H^1(F,T)\) for certain abelian extensions \(F/K\), with a certain relation between \(c_F\) and \(c_{F'}\) whenever \(F\subset F'\). The existence of an Euler system is used to bound Selmer groups attached to the Galois module \(\operatorname{Hom}(T,\mu_{p^{\infty}}\)). For example, Thaine’s results can be regarded as using cyclotomic units, interpreted as certain cohomology classes, to bound ideal class groups, interpreted as Selmer groups.

At present, very few Euler systems have been constructed. Besides the ones mentioned above, there is an important construction due to K. Kato for elliptic curves over \(\mathbb Q\). See the paper by A. Scholl [Galois representations in arithmetic algebraic geometry, London Math. Soc. Lect. Notes 254, 379-460 (1998; Zbl 0952.11015)]. A collection of cohomology classes, but not a full Euler system, has been used by M. Flach [Invent. Math. 109, 307-327 (1992; Zbl 0781.14022)] to bound the Selmer group of the symmetric square of an elliptic curve over \(\mathbb Q\).

The present book is a carefully written exposition of a major part of the theory of Euler systems. Chapter 1 discusses the cohomology of \(p\)-adic representations. and defines their Selmer groups. The duality theorems needed for future use are stated. Chapter 2 gives a definition of Euler systems and states the main results for bounding Selmer groups.

Chapter 3 gives the main concrete applications. Cyclotomic units are used to study the class groups of real abelian fields. Stickelberger elements are used to study the relative class groups of imaginary abelian fields. A proof is given for the main conjecture of Iwasawa theory, which was first proved by different techniques by B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)]. The present proof is essentially the one due to the author [Appendix to: S. Lang, Cyclotomic Fields I and II, Graduate Texts in Math. 121, Springer-Verlag, 397-419 (1990; Zbl 0704.11038)]. The Heegner point construction of Kolyvagin does not fit the author’s definition of an Euler system. However, the Euler system for the Tate module of a modular elliptic curve due to Kato does fit the definition, and it is discussed, with references to the literature for its constructions.

Chapters 4 through 7 are devoted to the proofs of the main theorems. Universal Euler systems, Kolyvagin’s derivative construction, twists of representations, and various aspects of Iwasawa theory are presented. Chapter 8 discusses the (mostly conjectural) relations between Euler systems and \(p\)-adic \(L\)-functions, especially those due to B. Perrin-Riou [Fonctions \(L\) \(p\)-adiques des représentations \(p\)-adiques, Astérisque 229 (1995; Zbl 0845.11040)]. Chapter 9 gives alternative definitions of Euler systems, including one that includes the construction using Heegner points.

There are four appendices giving necessary background results on topics such as Galois cohomology and \(p\)-adic representations.

The book does a nice job of collecting together and unifying many results from the literature. The author has been very careful to give precise references, either to results proved in the text or to results in the literature, when these are used in proofs. The subject by its nature is rather technical, and the general setting is even more so. The novice might profit from first reading the paper of Thaine cited above, perhaps a proof of the main conjecture of Iwasawa theory such as the one by the author [loc. cit.] or the exposition of the reviewer [Introduction to cyclotomic fields, 2nd edition, Graduate Texts in Math. 83, Springer-Verlag (1997; Zbl 0966.11047)], and one of the several discussions of Kolyvagin’s work, such as K. Rubin [Lect. Notes Math. 1399, 128-136 (1989; Zbl 0702.14028)] or B. Gross [\(L\)-functions and arithmetic (Durham, 1989), London Math. Soc. Lect. Notes 153, 235-256 (1991; Zbl 0743.14021)].

At present, very few Euler systems have been constructed. Besides the ones mentioned above, there is an important construction due to K. Kato for elliptic curves over \(\mathbb Q\). See the paper by A. Scholl [Galois representations in arithmetic algebraic geometry, London Math. Soc. Lect. Notes 254, 379-460 (1998; Zbl 0952.11015)]. A collection of cohomology classes, but not a full Euler system, has been used by M. Flach [Invent. Math. 109, 307-327 (1992; Zbl 0781.14022)] to bound the Selmer group of the symmetric square of an elliptic curve over \(\mathbb Q\).

The present book is a carefully written exposition of a major part of the theory of Euler systems. Chapter 1 discusses the cohomology of \(p\)-adic representations. and defines their Selmer groups. The duality theorems needed for future use are stated. Chapter 2 gives a definition of Euler systems and states the main results for bounding Selmer groups.

Chapter 3 gives the main concrete applications. Cyclotomic units are used to study the class groups of real abelian fields. Stickelberger elements are used to study the relative class groups of imaginary abelian fields. A proof is given for the main conjecture of Iwasawa theory, which was first proved by different techniques by B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)]. The present proof is essentially the one due to the author [Appendix to: S. Lang, Cyclotomic Fields I and II, Graduate Texts in Math. 121, Springer-Verlag, 397-419 (1990; Zbl 0704.11038)]. The Heegner point construction of Kolyvagin does not fit the author’s definition of an Euler system. However, the Euler system for the Tate module of a modular elliptic curve due to Kato does fit the definition, and it is discussed, with references to the literature for its constructions.

Chapters 4 through 7 are devoted to the proofs of the main theorems. Universal Euler systems, Kolyvagin’s derivative construction, twists of representations, and various aspects of Iwasawa theory are presented. Chapter 8 discusses the (mostly conjectural) relations between Euler systems and \(p\)-adic \(L\)-functions, especially those due to B. Perrin-Riou [Fonctions \(L\) \(p\)-adiques des représentations \(p\)-adiques, Astérisque 229 (1995; Zbl 0845.11040)]. Chapter 9 gives alternative definitions of Euler systems, including one that includes the construction using Heegner points.

There are four appendices giving necessary background results on topics such as Galois cohomology and \(p\)-adic representations.

The book does a nice job of collecting together and unifying many results from the literature. The author has been very careful to give precise references, either to results proved in the text or to results in the literature, when these are used in proofs. The subject by its nature is rather technical, and the general setting is even more so. The novice might profit from first reading the paper of Thaine cited above, perhaps a proof of the main conjecture of Iwasawa theory such as the one by the author [loc. cit.] or the exposition of the reviewer [Introduction to cyclotomic fields, 2nd edition, Graduate Texts in Math. 83, Springer-Verlag (1997; Zbl 0966.11047)], and one of the several discussions of Kolyvagin’s work, such as K. Rubin [Lect. Notes Math. 1399, 128-136 (1989; Zbl 0702.14028)] or B. Gross [\(L\)-functions and arithmetic (Durham, 1989), London Math. Soc. Lect. Notes 153, 235-256 (1991; Zbl 0743.14021)].

Reviewer: Lawrence Washington (College Park)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R34 | Galois cohomology |

11R23 | Iwasawa theory |

11R18 | Cyclotomic extensions |

11G05 | Elliptic curves over global fields |