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A generalized problem associated to the Kummer-Vandiver conjecture. (English) Zbl 1530.11086

For any number field \(F\) let \(A(F)\) be the \(p\)-part of the class group of \(F\), where \(p\) is a fixed odd prime. Whenever \(F\) is abelian over \(\mathbb Q\) (or more generally a CM field), the group \(A(F)\) splits as the direct sum \(A(F) = A(F)^+ \oplus A(F)^-\), and one can also identify \(A(F)^+\) with \(A(F^+)\), where \(F^+\) is the maximal real subfield of \(F\). The Kummer-Vandiver conjecture KVC (just called Vandiver conjecture by many authors) states that \(A(F)^+ = 0\) when \(F\) is the \(p\)-th cyclotomic field \(\mathbb Q(\zeta_p)\). In this case there is a finer decomposition of \(A(F)\), as the sum of subgroups \(A_{(k)}=e_{\omega^k}A(F)\) for \(0\le k < p-1\); the conjecture is equivalent to the vanishing of \(A_{(k)}\) for all even values of \(k\). The number of odd \(k\) for which \(A_{(k)}\) does not vanish is known as the irregularity index, and this can be positive; for example it is 1 for \(p=37\), the smallest irregular prime number. One can also define Iwasawa invariants \(\lambda(\omega^k)\) etc. attached to each character \(\omega^k\), and then KVC is equivalent to saying that \(\lambda(\omega^k)\), \(\mu(\omega_k)\) and and \(\nu(\omega^k)\) vanish for all even \(k\). The vanishing of all \(\mu's\) is a theorem in this setting, due to Ferrero and Washington.
While there are very plausible heuristics indicating that the Kummer-Vandiver conjecture may or should be true, a proof seems totally out of reach at present. The article under review takes a different stance: by using certain calculations that have a wider focus and which continue earlier work, lend credibility to the standard heuristics saying the number of exceptions to the conjecture with \(p<X\) should be at most \(\log \log X\).
To this end, \(F= \mathbb Q(\zeta_p)\) is replaced by \(K= \mathbb Q(\sqrt{d}, \zeta_p)\). That is, we now have twice as many relevant characters \(\psi = \chi^i \omega^k\) as before, where \(k=0,1\) and \(\chi\) is the nontrivial character of the quadratic field \(\mathbb Q(\sqrt{d})\); again half of these are even. To every \(\psi\) one can associate as before the Iwasawa invariants \(\lambda(\psi)\), \(\mu(\psi)\) and \(\nu(\psi)\), an Iwasawa series \(G_\psi(T)\), and an associated Iwasawa polynomial \(g_\psi(T)\) via Weierstraßpreparation. (Yet again all \(\mu(\psi)\) are zero by Ferrero-Washington.) If \(\psi\) is odd, then the degree of \(g_\psi(T)\) is \(\lambda(\psi)\). (For even \(\psi\), \(G_\psi\) and \(g_\psi\) are attached to a larger Iwasawa module, the \(p\)-ramified module.) The author now assumes \(\psi\) is even and defines four events depending on \(\psi\) and implicitly on \(p\): (1) \(\nu(\psi)>0\); (2) the constant coefficient of \(g_\psi\) is divisible by \(p^2\); (3) same as (2) replacing \(g_\psi\) by its Kummer twist \(g^*_\psi\); and finally (3) \(g_\psi\) (or equivalently \(g^*_\psi\)) has degree larger than 1. He then calls the pair \((p,\psi)\) exceptional if one of these four events happens. This definition is not new, appearing already in [H. Sumida-Takahashi, J. Number Theory 105, No. 2, 235–250 (2004; Zbl 1048.11085)] (see below).
The paper formulates a conjecture (Conj. 3.1) that intends to asymptotically predict the number of exceptional pairs, un-fixing \(p\), and even sorted by the four types of events, and the paper also provides very ample numerical evidence for this conjecture, visualized by coloured diagrams. In the last section, the author sketches the underlying computations, referring to earlier computations which followed quite similar goals and were described in four earlier papers from the years 2004–2007, numbers [11] to [14] in the references [H. Sumida-Takahashi, J. Number Theory 105, No. 2, 235–250 (2004; Zbl 1048.11085); Exp. Math. 14, No. 3, 307–316 (2005; Zbl 1082.11071; Math. Comput. 76, No. 258, 1059–1071 (2007; Zbl 1192.11076); J. Math., Univ. Tokushima 41, 33–41 (2007; Zbl 1145.11075)]. The setting and the style of the present paper are closely related to [11]–[14], and the explanations about the ideas behind the calculation and the details of the approach are not self-contained; one should go back to the earlier papers for a deeper understanding. A central point seems to be the following. One may bound class numbers in the plus part, by showing that certain cyclotomic units are not \(p^e\)-th powers for suitable \(e\), and this can be done by a local argument. For the converse, that is, for showing that a cyclotomic unit is actually a high power, the author has invented a clever method using Gauss sums, already in the first of the four mentioned papers. That method seems more efficient than a direct attack using minimal polynomials, used by J. S. Kraft and R. Schoof [Compos. Math. 97, No. 1–2, 135–155 (1995; Zbl 0840.11043)]). The objects that need to be calculated are fourfold: beyond cyclotomic units and Gauss sums one also wants generalized Bernoulli numbers and Iwasawa polynomials. The latter can only be determined modulo a suitably high \(p\)-power. The precise link to KVC as it stands is not explained in detail and not completely clear. When we switch the quadratic field \(\mathbb Q(\sqrt{d})\) back to \(\mathbb Q\), then assuming Greenberg’s conjecture, KVC holds iff event (1) happens for no even character \(\omega^k\). One can say more, see the second sentence in Nguyen Quang Do’s review MR2380210 of [H. Sumida-Takahashi, J. Math., Univ. Tokushima 41, 33–41 (2007; Zbl 1145.11075)]. – In conclusion it should be mentioned that it took many years of work and computer calculations to assemble the numerical results presented in this paper, and their ampleness is impressive.

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
Full Text: DOI

References:

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