Stickelberger elements and Kolyvagin systems. (English) Zbl 1257.11099

Büyükboduk, Kâzım

Nagoya Math. J. 203, 123-173 (2011).

Let \(k\) be a totally real field with absolute Galois group \(G_k\) and let \(p\) be an odd prime. Fix an odd character \(\chi: G_k \rightarrow \mathbb Z_p^{\times}\) different from the Teichmüller character and a field \(K\) containing the fixed field \(L\) cut out by \(\chi\). The author assumes that the \(\chi\)-part of the \(p\)-part of Brumer’s conjecture holds for the extension \(K/k\), i.e. the \(\chi\)-part of the Stickelberger element \(\theta_K^{\chi}\) annihilates \(A_K^{\chi}\), the \(\chi\)-isotypic part of the \(p\)-part of the class group of \(K\). He then uses the Euler system / Kolyvagin system machinery to prove the equality \[ |A_L^{\chi}| = |\mathbb Z_p / \theta_L^{\chi} \mathbb Z_p| \] under the condition that \(\chi(\wp) \not=1\) for any prime \(\wp\) of \(k\) above \(p\). With a bit more work he then derives from this a variant of the main conjecture, namely he shows that the characteristic polynomial of the Pontryagin dual of the direct limit of the \(A_{L_n}^{\chi}\) (where \(L_n\) denotes the \(n\)-th layer in the cyclotomic tower) is given by a certain twist of the Deligne-Ribet \(p\)-adic \(L\)-function. This is done under the additional assumption that any prime of \(k\) above \(p\) totally ramifies in the cyclotomic extension (reviewer’s remark: on top of p. 125 the author mentions that this is true whenever \(k/\mathbb Q\) is unramified; but what he certainly means is that \(p\) does not ramify – otherwise this would not provide many examples). Note that as a consequence of the above condition on \(\chi\), the associated \(p\)-adic \(L\)-series has no trivial zeros.
We point out that none of these results is really new; in fact both have been proven unconditionally and in greater generality by A. Wiles [Ann. Math. (2) 131, No. 3, 493–540 (1990; Zbl 0719.11071) and Ann. Math. (2) 131, No. 3, 555–565 (1990; Zbl 0719.11082)]. But the purpose of this article is to explore how Kolyvagin systems may be used to bound the size of a Selmer group when the core Selmer rank (in the sense of B. Mazur and K. Rubin [Kolyvagin systems. Mem. Am. Math. Soc. 799, 96 p. (2004; Zbl 1055.11041)]) is strictly greater than \(1\). Here, the core Selmer rank of the considered Galois representations equals the degree \([k: \mathbb Q]\).
Finally, note that the author first shows one inequality of the above displayed equality and then uses the usual ‘class number trick’ in order to obtain equality. For this, he assumes that this inequality holds for the Teichmüller character as well (if a \(p\)-th root of unity belongs to \(L\)). But in fact, one needs this inequality for all characters of \(\text{Gal}(L/k)\), some of which may fail the above condition. The author kindly explained me that one still can prove the corresponding statement for all the remaining characters by considering an ‘improved Euler system’; the argument is then very similar to what happens in the author’s article [J. Reine Angew. Math. 631, 85–107 (2009; Zbl 1216.11102)].

Reviewer: Andreas Nickel (Bielefeld)

MSC:

11R23	Iwasawa theory
11R42	Zeta functions and \(L\)-functions of number fields
11R29	Class numbers, class groups, discriminants

Keywords:

Stickelberger elements; Kolyvagin systems; Euler systems; class groups; main conjecture; Iwasawa theory

Citations:

Zbl 0719.11071; Zbl 0719.11082; Zbl 1055.11041; Zbl 1216.11102

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