History. In 1986, Francisco Thaine [Th] introduced a remarkable method for bounding ideal class groups of real abelian extensions of Q. Namely, ifFis such a field, he used cyclotomic units in fieldsF(μ_ℓ), for a large class of rational primesℓ, to construct explicitly a large collection of principal ideals ofF. His construction produced enough principal ideals to bound the exponent of the different Galois-eigencomponents of the ideal class group ofF, in terms of the cyclotomic units ofF. Although Thaine’s results were already known as a corollary of Iwasawa’s “main conjecture”, proved by Mazur and...

In this chapter we introduce our basic objects of study:p-adic Galois representations, their cohomology groups, and especially Selmer groups.

We begin by recalling basic facts about cohomology groups associated top-adic representations, material which is mostly well-known but included here for completeness.

A Selmer group is a subgroup of a global cohomology group determined by “local conditions”. In §1.3 we discuss these local conditions, which are defined in terms of special subgroups of the local cohomology groups. In §1.4 we state without proof the results we need concerning the Tate pairing on local cohomology groups, and we study how...

In this chapter we state our main results. The definition of an Euler system is given in §2.1, and the theorems applying Euler systems to study Selmer groups over number fields and over$Z_{p}^{d}$-extensions of number fields are given in §2.2 and §2.3, respectively. Examples and applications are given in Chapter 3; the reader might benefit from following along in those examples while reading this chapter. The proofs, using tools to be developed in Chapter 4, will be given in Chapters 5 and 7. In Chapter 9 we discuss some variants and extensions of the definition of Euler system...

In this chapter we give the basic examples of Euler systems and their applications, using the results of Chapter 2.

Supposeχis a character ofG_Kinto$\cal{O}^x$. As in Example 1.1.2 we will denote by$\cal{O}_x$a free rank-one$\cal{O}$-module on whichG_Kacts viaχ. Recall that D = Φ/$\cal{O}$=$\cal{O}$⊗ (Q_p/Z_p)· We will also write

D_x= D ⊗_o$\cal{O}_x$=$\cal{O}_x$⊗ (Q_p/Z_p)·

For the first three examples (§§3.2, 3.3, and 3.4) we will assume thatχhas finite prime-to-porder. As in §1.6.B and §1.6.C we takeT=...

The proofs of the main theorems stated in Chapter 2 consist of two steps. First we use an Euler system to construct auxiliary cohomology classes which Kolyvagin calls “derivative” classes, and second we use these derived classes along with the duality theorems of § 1.7 to bound Selmer groups.

In this chapter we carry out the first of these steps. In §4.2 and §4.3 we define and study the “universal Euler system” associated toTandK_∞/K. In §4.4 we construct the Kolyvagin derivative classes, and in §4.5 we state the local properties of these derivative classes, which will be crucial...

In this chapter we will prove Theorems 2.2.2 (in §5.2) and 2.2.3 (in §5.3). For every powerMofpwe will choose inductively a finite subset ∑ of primes inR_K,M· As τ runs through products of primes in ∑, Theorem 4.5.1 shows that the derivative cohomology classesK_[K,τ,M]defined in Chapter 4 belong toS^∑U∑p(K, W_M), where ∑pis the set of primes ofKabovep, and Theorem 4.5.4 tells us about the singular parts of these classes at primes in ∑. This information and the duality results of §1.7 will allow us to bound the index...

In this chapter we extend the methods of §2.4 to twist Euler systems by characters of infinite order. This will be used in Chapter 7 when we prove Theorems 2.3.2, 2.3.3, and 2.3.4. Ifpis a character of Gal(K_∞/K), then

Theorem 6.3.5 says that an Euler system c for (T,K_∞) gives rise to an Euler system c^pfor (T⊗p,K_∞),

Theorem 6.4.1 shows that the theorems of §2.3 hold forTand c if and only if they hold forT⊗pand c^p, and

Lemma 6.1.3 allows us to choose a particularpwhich...

In this chapter we use the cohomology classes constructed in Chapter 4, along with the duality results of §1.7, to prove Theorems 2.3.2, 2.3.3, and 2.3.4. The proofs follow generally along the same lines as the proof of Theorem 2.2.2 given in Chapter 5, except that where in Chapter 5 we dealt with$\cal{O}$-modules, we must now deal with$\cal{O}$[Gal(F/K)]-modules forK ⊂_tF ⊂ K_∞. This makes the algebra much more complicated.

In §7.1 we give the proofs of Theorems 2.3.3 and 2.3.4, assuming Theorem 2.3.2 and two propositions (Propositions 7.1.7 and 7.1.9), whose proofs will be...

So far we have discussed at length how an Euler system for ap-adic representationTofG_Kcontrols the Selmer groupsS(K, W*) andS(K_∞, W*). This raises several natural questions which we have not yet touched on.

Except for the examples in Chapter 3, we have not discussed at all how to produce Euler systems. For which representations do (nontrivial) Euler systems exist?

If there is a nontrivial Euler system c forT, then there are infinitely many such (for example, we can act on c by elements of %$\cal{O}[[G_K]]). Is there a “best” Euler system?

Conjecturally,...

In this chapter we discuss several alternatives and extensions to the definition of Euler systems we gave in Chapter 2.

It is tempting to remove from the definition of an Euler system the requirement that the fieldK(over whose subfields the Euler system classes are defined) contain a Z_p-extension ofK. After all, the proofs of the Theorems of §2.2 only use the derivative classesK_[K,τ,M]and not theK_[F,τ,M]for larger extensionsFofKinK_∞. However, our proofs of the properties of the derivative classesK_[K,τ,M]very much used the fact that the Euler system class...