##
**Kolyvagin systems of Stark units.**
*(English)*
Zbl 1216.11102

Fix a totally real number field \(k\) of degree \(r\) and an odd prime \(p\). The paper under review is devoted to an explicit construction, based upon the conjectural existence of Stark elements proposed in [K. Rubin, “A Stark conjecture ‘over \(\mathbb Z\)’ for abelian \(L\)-functions with multiple zeros”, Ann. Inst. Fourier 46, No. 1, 33–62 (1996; Zbl 0834.11044)], of a modified Selmer structure for the representation \(\mathbb{Z}_p(1)\otimes\chi\) of an absolute Galois group of \(k\), where \(\chi\) is an even, non-trivial character unramified outside of \(p\). The main application of this construction is a generalization of Gras’ Conjecture, namely a formula connecting orders of \(\chi\)-components of, on a one side, class groups of abelian extensions of \(k\); and, on the other, quotients of exterior powers of full groups of units modulo the modules generated by Stark elements.

Section 1 is concerned with a refined definition of Selmer structure, which is mainly based on the classical one of Mazur-Rubin for the class group as in their book [B. Mazur and K. Rubin, Kolyvagin systems. Mem. Am. Math. Soc. 168, No. 799 (2004; Zbl 1055.11041)] but insisting on a stricter condition at \(p\), thus only considering one line \(\mathcal{L}\) instead of the whole local cohomology group with \(\mathbb{Z}_p(1)\otimes\chi\)-coefficients. The main result of this section is Corollary 1.9 which shows the one-dimensionality of Kolyvagin Systems relative to this stricter condition (as in Theorem 5.2.10 of Mazur-Rubin’s book).

In Section 2 the author recalls the definition, due to Rubin, of certain Stark elements

\[ \varepsilon_{K,S} \in \bigwedge^{r-1} \mathcal{O}_{K,S}^{\times,\land} \] where \(S\) is a finite set of allowed ramification and \(K\) can vary through all abelian extensions of \(k\) unramified outside of primes depending on \(p\) and \(\chi\). The author describes how to produce, out of a collection of Stark elements, an Euler system for \(\mathbb{Z}_p(1)\otimes \chi^{-1}\) for every multi-linear map \[ \Phi\in\varprojlim_K\bigwedge^{r-1}\mathrm{Hom}_{\mathbb{Z}_p[\Delta_K]}(V_K,\mathbb{Z}[\Delta_K]) \] where, for \(K\) as above, \(\Delta_K\) denotes \(\mathrm{Gal}(K/k)\) and \(V_K\) is the \(p\)-adic completion of \(\mathcal{O}_K^\times\subseteq K\otimes\mathbb{Q}_p\) – this is Proposition 2.8. In the last part of Section 2 a concrete instance of a Kolyvagin system of the form of those discussed in Section 1 is presented: the author starts by fixing a line \(\mathbb{L}\subseteq \mathbb{V}\) inside the projective limit of the completions of units in all \(K\)’s which fullfils some property (see after Corollary 2.13) but is not unique, in general (there is a canonical choice for such an \(\mathbb{L}\) if one prime of \(k\) above \(p\) is totally split). This is the fundamental choice upon which the rest of the paper will depend. Denote by \(\mathcal{L}_K\) be the image of \(\mathbb{L}\) inside \(V_K\). Proposition 2.17 gives a concrete construction of a multi-linear \(\Phi\) as above that, moreover, takes values in \(\mathbb{L}\). It is then easy (Theorem 2.19) to show that applying the map introduced in Theorem 3.2.4 of Mazur-Rubin’s book to the Euler system generated with such a \(\Phi\) produces a Kolyvagin System relative to the stricter condition defined by \(\mathcal{L}_K\) in the sense of Section 1. It is crucial to stress that the whole construction can be performed not only at the level of \(k\) but also character-wise for abelian extensions of \(L=\overline{k}^{\mathrm{ker}(\chi)}\).

Section 3 draws the conclusions of the above discussion by applying the machinery introduced in Mazur-Rubin’s book [loc. cit.] to the Kolyvagin system produced at the end of Section 2. The main application is Theorem 3.10 which bounds the \(\chi\)-component of the class group by the index of the Euler system in the exterior product \(\bigwedge^r(\mathcal{O}_L^\times)^\chi\): this bound, together with a careful application of the analytic class number formula gives the following

Theorem 3.11 Let \(L\) be the fixed field of \(\mathrm{ker}(\chi)\) in an algebraic closure of \(k\). Let \(A_L^\chi\) be the \(\chi\)-component of the \(p\)-part of the class group of \(L\) and assume the existence of the “Stark element” \(\varepsilon_L\) of Rubin. If the Leopoldt Conjecture holds for \(L\) and \(p\), then \[ |A_L^\chi|=\left[\bigwedge^r(\mathcal{O}_L^\times)^\chi:\mathbb{Z}_p\varepsilon_L^\chi\right]\;. \] Corollary 3.12 With notations as in Theorem 3.11 one has \[ \lim_{s\rightarrow 0} s^{-r}L_{S_\infty,T}(s,\chi^{-1})\sim \mathcal{R}_L^\chi\cdot| A_L^\chi| \] where \(\sim\) means “up to \(p\)-adic units”, \(L_{S_\infty,T}\) is the modified Dirichlet \(L\)-function as defined for instance in Section 1.1 of [K. Rubin, “A Stark conjecture “over \(\mathbb{Z}\)” for abelian \(L\)-functions with multiple zeros,” Ann. Inst. Fourier 46, No.1, 33–62 (1996; Zbl 0834.11044)] and \(\mathcal{R}_L^\chi\) is the value of a certain regulator map introduced in [ibid.].

Section 1 is concerned with a refined definition of Selmer structure, which is mainly based on the classical one of Mazur-Rubin for the class group as in their book [B. Mazur and K. Rubin, Kolyvagin systems. Mem. Am. Math. Soc. 168, No. 799 (2004; Zbl 1055.11041)] but insisting on a stricter condition at \(p\), thus only considering one line \(\mathcal{L}\) instead of the whole local cohomology group with \(\mathbb{Z}_p(1)\otimes\chi\)-coefficients. The main result of this section is Corollary 1.9 which shows the one-dimensionality of Kolyvagin Systems relative to this stricter condition (as in Theorem 5.2.10 of Mazur-Rubin’s book).

In Section 2 the author recalls the definition, due to Rubin, of certain Stark elements

\[ \varepsilon_{K,S} \in \bigwedge^{r-1} \mathcal{O}_{K,S}^{\times,\land} \] where \(S\) is a finite set of allowed ramification and \(K\) can vary through all abelian extensions of \(k\) unramified outside of primes depending on \(p\) and \(\chi\). The author describes how to produce, out of a collection of Stark elements, an Euler system for \(\mathbb{Z}_p(1)\otimes \chi^{-1}\) for every multi-linear map \[ \Phi\in\varprojlim_K\bigwedge^{r-1}\mathrm{Hom}_{\mathbb{Z}_p[\Delta_K]}(V_K,\mathbb{Z}[\Delta_K]) \] where, for \(K\) as above, \(\Delta_K\) denotes \(\mathrm{Gal}(K/k)\) and \(V_K\) is the \(p\)-adic completion of \(\mathcal{O}_K^\times\subseteq K\otimes\mathbb{Q}_p\) – this is Proposition 2.8. In the last part of Section 2 a concrete instance of a Kolyvagin system of the form of those discussed in Section 1 is presented: the author starts by fixing a line \(\mathbb{L}\subseteq \mathbb{V}\) inside the projective limit of the completions of units in all \(K\)’s which fullfils some property (see after Corollary 2.13) but is not unique, in general (there is a canonical choice for such an \(\mathbb{L}\) if one prime of \(k\) above \(p\) is totally split). This is the fundamental choice upon which the rest of the paper will depend. Denote by \(\mathcal{L}_K\) be the image of \(\mathbb{L}\) inside \(V_K\). Proposition 2.17 gives a concrete construction of a multi-linear \(\Phi\) as above that, moreover, takes values in \(\mathbb{L}\). It is then easy (Theorem 2.19) to show that applying the map introduced in Theorem 3.2.4 of Mazur-Rubin’s book to the Euler system generated with such a \(\Phi\) produces a Kolyvagin System relative to the stricter condition defined by \(\mathcal{L}_K\) in the sense of Section 1. It is crucial to stress that the whole construction can be performed not only at the level of \(k\) but also character-wise for abelian extensions of \(L=\overline{k}^{\mathrm{ker}(\chi)}\).

Section 3 draws the conclusions of the above discussion by applying the machinery introduced in Mazur-Rubin’s book [loc. cit.] to the Kolyvagin system produced at the end of Section 2. The main application is Theorem 3.10 which bounds the \(\chi\)-component of the class group by the index of the Euler system in the exterior product \(\bigwedge^r(\mathcal{O}_L^\times)^\chi\): this bound, together with a careful application of the analytic class number formula gives the following

Theorem 3.11 Let \(L\) be the fixed field of \(\mathrm{ker}(\chi)\) in an algebraic closure of \(k\). Let \(A_L^\chi\) be the \(\chi\)-component of the \(p\)-part of the class group of \(L\) and assume the existence of the “Stark element” \(\varepsilon_L\) of Rubin. If the Leopoldt Conjecture holds for \(L\) and \(p\), then \[ |A_L^\chi|=\left[\bigwedge^r(\mathcal{O}_L^\times)^\chi:\mathbb{Z}_p\varepsilon_L^\chi\right]\;. \] Corollary 3.12 With notations as in Theorem 3.11 one has \[ \lim_{s\rightarrow 0} s^{-r}L_{S_\infty,T}(s,\chi^{-1})\sim \mathcal{R}_L^\chi\cdot| A_L^\chi| \] where \(\sim\) means “up to \(p\)-adic units”, \(L_{S_\infty,T}\) is the modified Dirichlet \(L\)-function as defined for instance in Section 1.1 of [K. Rubin, “A Stark conjecture “over \(\mathbb{Z}\)” for abelian \(L\)-functions with multiple zeros,” Ann. Inst. Fourier 46, No.1, 33–62 (1996; Zbl 0834.11044)] and \(\mathcal{R}_L^\chi\) is the value of a certain regulator map introduced in [ibid.].

Reviewer: Filippo Nuccio (Heidelberg)

### MSC:

11R32 | Galois theory |

11R23 | Iwasawa theory |

11R42 | Zeta functions and \(L\)-functions of number fields |

### References:

[1] | Benedict, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1) pp 177– (1988) |

[2] | Krasner Marc, Acta Arith. 3 pp 133– (1939) |

[3] | Mazur Barry, Mem. Amer. Math. Soc. 168 (799) pp 96– (2004) |

[4] | Perrin-Riou Bernadette, Ann. Inst. Fourier (Grenoble) 48 (5) pp 1231– (1998) |

[5] | DOI: 10.1023/A:1001586625441 · Zbl 0939.11037 · doi:10.1023/A:1001586625441 |

[6] | Rubin Karl, Math. 425 pp 141– (1992) |

[7] | Solomon David, Ann. Inst. Fourier (Grenoble) 52 (2) pp 379– (2002) |

[8] | DOI: 10.1006/jnth.2001.2724 · Zbl 1032.11051 · doi:10.1006/jnth.2001.2724 |

[9] | DOI: 10.2307/2118559 · Zbl 0823.11029 · doi:10.2307/2118559 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.