##
**Kolyvagin systems.**
*(English)*
Zbl 1055.11041

Mem. Am. Math. Soc. 168, No. 799, viii, 96 p. (2004).

The theory of Kolyvagin systems is a recent outgrowth of the theory of Euler systems. The latter has its roots in work of Thaine and Kolyvagin; it was developed to full bloom by Kolyvagin and Rubin. For some more details on history, we refer to L. Washington’s review [“Euler systems. (Hermann Weyl Lectures)”. (Ann. Math. Stud. 147. Princeton, NJ: Princeton University Press) (2000; Zbl 0977.11001)] of K. Rubin’s monograph on Euler systems; for a detailed description of the theory we refer to that monograph itself. We briefly recall that an Euler system is a collection of cohomology classes \(c_F\in\text{H}^1(F,T)\) where \(T\) is a \(p\)-adic Galois representation over a number field \(K\), \(F\) runs through a suitable infinite family of Abelian extensions of \(K\), and the \(c_F\) have to satisfy certain compatibility relations involving Euler factors. The most classical example has \(T=\mu_{p^n}\) and the classes \(c_F\) are directly furnished by cyclotomic units. The raison d’être for such Euler systems is that they produce bounds on the size of the first cohomology of the Tate dual of \(T\). In the most classical example this amounts simply to \(p\)-parts of class groups. We note that it is a lot easier to get annihilators than bounds on the size; it is the latter task which requires the full works, which we now sketch.

Given an Euler system, one first passes to a related system of classes \(\kappa_F\) which now are in \(\text{H}^1(K,T)\) (i.e., defined over the base field); what varies now is a set of local conditions, which become more and more relaxed as \(F\) grows. The very nice idea is now to consider this system of “Kolyvagin derivatives” in its own right, deliberately forgetting that it comes from an Euler system. This is justified by the fact that its properties, properly axiomatised, allow to prove the same bounds as previously. As the authors point out in their very well-written introduction, these new structures, which are aptly named Kolyvagin systems, exhibit a rigidity not shared by Euler systems.

A little more precisely: A Kolyvagin system is defined as a global section of a certain sheaf; this sheaf is a functor from a graph \(\mathcal N\) to \(R\)-modules. The graph \(\mathcal N\) has for vertices a set of squarefree natural numbers \(n\) satisfying extra conditions, with edges from \(n\) to \(n\ell\) whenever \(\ell\) is prime and \(n,n\ell\in\mathcal N\). To every vertex one attaches a group \(\text{H}^1_n(K,T)\), where the \(n\) stands for local conditions that get more relaxed as \(n\) grows. The most standard example would be \(R={\mathbb Z}/M\), \(T=\mu_{M}\). Then \(\text{H}^1(K,T)\) without any local conditions is \(K^*/{K^*}^M\), and in the classical cyclotomic example, \(\kappa_n\) is indeed in \(K^*/{K^*}^M\), but with the extra property that the valuation is zero mod \(M\) at all places not dividing \(n\). The connection between the Kolyvagin elements \(\kappa_n\) and \(\kappa_{n\ell}\) is axiomatized in a very clever manner: the map \(\varphi_\ell\) (p. 401 of K. Rubin’s appendix to S. Lang’s “Cyclotomic Fields” [second edition, Springer GTM 121 (1990; Zbl 0704.11038)]) is replaced by the so-called “finite-to-singular” map defined entirely in terms of cohomology. (We over-simplify here; in fact on also has to attach \(R\)-modules to the edges to the graph, in order to make this work.) To make that map entirely canonical, one tensors its domain and target by an appropriate Galois group, so the choice of generators that was implicit in \(\varphi_\ell\) drops out. In an appendix it is proved that Euler system do lead to a Kolyvagin system as it should be. (In Chapter 4 of Rubin’s book on Euler systems one obtains what is called “weak Kolyvagin systems” from the new perspective. For a detailed example showing the difference see Example 3.1.10 and note the difference between \({\mathcal F}(n)\) and \({\mathcal F}^n\), the latter being strictly larger in general.)

In Chapters 4 and 5, the central parts of the theory are developed, first for uniserial Artin rings \(R\), then for DVR’s (prime example \(R={\mathbb Z}_p\)) and finally for \(R=\Lambda\). As said before, Kolyvagin systems can be made more rigid than Euler systems, and this is achieved by putting them into smaller modules. First, as just explained, the local conditions at vertex \(n\) are somewhat stricter than those worked with previously (“weak Kolyvagin systems”). Second, it turns out one can make the vertex modules even smaller by considering the so-called stub submodules inside them. Very roughly speaking, one forces to have the vertex modules the “expected rank” \(\chi(T)\), the so-called core rank of the Kolyvagin system, and then under suitable hypotheses all elements of the Kolyvagin system already lie in the stub submodule, which gives precise results about the module of all Kolyvagin systems. Indeed one then obtains results which say that the Fitting ideal of the dual Selmer group is completely determined by the set of all elements that start a Kolyvagin system (see 0.2 and 0.3 of introduction, Thm. 5.2.14, Thm. 5.3.10(iii)). It is very hard to give an idea of the technical core of the proofs in a review; let us just say that duality theorems and the judicious choice of auxiliary primes still play a big role (the reader might have a look at the proof of Cor. 4.1.9 and all that goes into it), as on earlier occasions. In the comparatively short sixth chapter, the authors show how the preceding theory leads to results for the typical examples (cyclotomic theory, elliptic curves). It turns out that in the cyclotomic setting, the standard Kolyvagin system is primitive (roughly speaking, not divisible), whereas Kato’s Kolyvagin system need not be primitive.

There are two appendices, each containing the long and technical proof of a result that is needed in the main text: in (A) the statement that Euler systems do lead to Kolyvagin systems, and in (B) an important theorem, due to B. Howard, concerning stub Selmer modules.

This work represents quite important progress, which should result in a lot of follow-up work, and is highly rewarding reading. It is only natural that the level of abstraction and complexity goes up another notch with respect to the established theory of Euler systems. As the authors recommend themselves, it is helpful to look at the final chapter early on, in order to see examples. Some important applications of Euler systems are retrieved there: the Gras Conjecture (= theorem of Mazur and Wiles) and various Main Conjectures.

The reviewer found very few misprints; here they are, for the reader’s convenience. In Example 2.3.2, the indices \(a\) and \(b\) on \(\mathcal F^*\) should be exchanged. On p.15 (middle), second bullet, first case, read \(l \nmid abc\). In 6.1.10, \(\kappa^{\text{Kato}}\) should presumably read \(\kappa^\rho\). Finally, the statement of Lemma 2.1.4 is not true in full generality. If however \(T/\mathbf{m}T\) is assumed to be irreducible, then all is safe, and this is the only case needed from Chapter 3 onward (note that in 3.5.2 Hypothesis (H.1) should be assumed exactly as in 3.5.3). The reviewer would like to thank Karl Rubin for a very helpful email reply concerning this last issue.\)

Given an Euler system, one first passes to a related system of classes \(\kappa_F\) which now are in \(\text{H}^1(K,T)\) (i.e., defined over the base field); what varies now is a set of local conditions, which become more and more relaxed as \(F\) grows. The very nice idea is now to consider this system of “Kolyvagin derivatives” in its own right, deliberately forgetting that it comes from an Euler system. This is justified by the fact that its properties, properly axiomatised, allow to prove the same bounds as previously. As the authors point out in their very well-written introduction, these new structures, which are aptly named Kolyvagin systems, exhibit a rigidity not shared by Euler systems.

A little more precisely: A Kolyvagin system is defined as a global section of a certain sheaf; this sheaf is a functor from a graph \(\mathcal N\) to \(R\)-modules. The graph \(\mathcal N\) has for vertices a set of squarefree natural numbers \(n\) satisfying extra conditions, with edges from \(n\) to \(n\ell\) whenever \(\ell\) is prime and \(n,n\ell\in\mathcal N\). To every vertex one attaches a group \(\text{H}^1_n(K,T)\), where the \(n\) stands for local conditions that get more relaxed as \(n\) grows. The most standard example would be \(R={\mathbb Z}/M\), \(T=\mu_{M}\). Then \(\text{H}^1(K,T)\) without any local conditions is \(K^*/{K^*}^M\), and in the classical cyclotomic example, \(\kappa_n\) is indeed in \(K^*/{K^*}^M\), but with the extra property that the valuation is zero mod \(M\) at all places not dividing \(n\). The connection between the Kolyvagin elements \(\kappa_n\) and \(\kappa_{n\ell}\) is axiomatized in a very clever manner: the map \(\varphi_\ell\) (p. 401 of K. Rubin’s appendix to S. Lang’s “Cyclotomic Fields” [second edition, Springer GTM 121 (1990; Zbl 0704.11038)]) is replaced by the so-called “finite-to-singular” map defined entirely in terms of cohomology. (We over-simplify here; in fact on also has to attach \(R\)-modules to the edges to the graph, in order to make this work.) To make that map entirely canonical, one tensors its domain and target by an appropriate Galois group, so the choice of generators that was implicit in \(\varphi_\ell\) drops out. In an appendix it is proved that Euler system do lead to a Kolyvagin system as it should be. (In Chapter 4 of Rubin’s book on Euler systems one obtains what is called “weak Kolyvagin systems” from the new perspective. For a detailed example showing the difference see Example 3.1.10 and note the difference between \({\mathcal F}(n)\) and \({\mathcal F}^n\), the latter being strictly larger in general.)

In Chapters 4 and 5, the central parts of the theory are developed, first for uniserial Artin rings \(R\), then for DVR’s (prime example \(R={\mathbb Z}_p\)) and finally for \(R=\Lambda\). As said before, Kolyvagin systems can be made more rigid than Euler systems, and this is achieved by putting them into smaller modules. First, as just explained, the local conditions at vertex \(n\) are somewhat stricter than those worked with previously (“weak Kolyvagin systems”). Second, it turns out one can make the vertex modules even smaller by considering the so-called stub submodules inside them. Very roughly speaking, one forces to have the vertex modules the “expected rank” \(\chi(T)\), the so-called core rank of the Kolyvagin system, and then under suitable hypotheses all elements of the Kolyvagin system already lie in the stub submodule, which gives precise results about the module of all Kolyvagin systems. Indeed one then obtains results which say that the Fitting ideal of the dual Selmer group is completely determined by the set of all elements that start a Kolyvagin system (see 0.2 and 0.3 of introduction, Thm. 5.2.14, Thm. 5.3.10(iii)). It is very hard to give an idea of the technical core of the proofs in a review; let us just say that duality theorems and the judicious choice of auxiliary primes still play a big role (the reader might have a look at the proof of Cor. 4.1.9 and all that goes into it), as on earlier occasions. In the comparatively short sixth chapter, the authors show how the preceding theory leads to results for the typical examples (cyclotomic theory, elliptic curves). It turns out that in the cyclotomic setting, the standard Kolyvagin system is primitive (roughly speaking, not divisible), whereas Kato’s Kolyvagin system need not be primitive.

There are two appendices, each containing the long and technical proof of a result that is needed in the main text: in (A) the statement that Euler systems do lead to Kolyvagin systems, and in (B) an important theorem, due to B. Howard, concerning stub Selmer modules.

This work represents quite important progress, which should result in a lot of follow-up work, and is highly rewarding reading. It is only natural that the level of abstraction and complexity goes up another notch with respect to the established theory of Euler systems. As the authors recommend themselves, it is helpful to look at the final chapter early on, in order to see examples. Some important applications of Euler systems are retrieved there: the Gras Conjecture (= theorem of Mazur and Wiles) and various Main Conjectures.

The reviewer found very few misprints; here they are, for the reader’s convenience. In Example 2.3.2, the indices \(a\) and \(b\) on \(\mathcal F^*\) should be exchanged. On p.15 (middle), second bullet, first case, read \(l \nmid abc\). In 6.1.10, \(\kappa^{\text{Kato}}\) should presumably read \(\kappa^\rho\). Finally, the statement of Lemma 2.1.4 is not true in full generality. If however \(T/\mathbf{m}T\) is assumed to be irreducible, then all is safe, and this is the only case needed from Chapter 3 onward (note that in 3.5.2 Hypothesis (H.1) should be assumed exactly as in 3.5.3). The reviewer would like to thank Karl Rubin for a very helpful email reply concerning this last issue.\)

Reviewer: Cornelius Greither (Neubiberg)