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Mahler measure and Weber’s class number problem in the cyclotomic \(\mathbb Z_p\)-extension of \(\mathbb Q\) for odd prime number \(p\). (English) Zbl 1280.11069

Let \(p\) be a prime number, \(\mu_m\) the group of all \(m\)-th roots of unity in \(\mathbb C\) and \(\mathbb Q(\mu_{ p^{\infty}}):=\bigcup_{n\geq 1}\mathbb Q(\mu_{p^n})\). Let \(\mathbf B_{p,\infty}\) be the unique real subfield of \(\mathbb Q(\mu_{p^\infty})\) whose Galois group \(\text{Gal}(\mathbf B_{p^\infty}/\mathbb Q)\) is topologically isomorphic to the \(p\)-adic integers ring \(\mathbf Z_p\) as additive groups. Let \(\mathbf B_{p,n}\) be the unique subfield of \(\mathbf B_{p,\infty}\) which is cyclic of degree \(p^n\) over \(\mathbb Q\) and \(h_{p,n}\) its class number.
The computation of \(h_{p,n}\) is very difficult for large \(p^n\). An approach to try to lessen this difficulty is to answer the problem: Let \(l\) be a prime number. Does \(l\) divide \(h_{p,n}\) for any positive integer \(n\)? (this problem generalizing Weber’s class number problem: Is the class number \(h_{p,n}\) equal to one for any positive integer \(n\)?).
Towards this problem K. Horie and M. Horie developed a method for proving \(l\)-indivisibility of \(h_{p,n}\) under certain conditions see [K. Horie, J. Lond. Math. Soc., II. Ser. 66, No. 2, 257–275 (2002; Zbl 1011.11072); Proc. Japan Acad., Ser. A 81, No. 3, 40–43 (2005; Zbl 1114.11086); Tohoku Math. J. (2) 57, No. 3, 375–394 (2005; Zbl 1128.11051); Tohoku Math. J. (2) 59, No. 2, 259–291 (2007; Zbl 1202.11050)] and [K. Horie and M. Horie, Acta Arith. 135, No. 2, 159–180 (2008; Zbl 1158.11046); Tohoku Math. J. (2) 61, No. 4, 551–570 (2009; Zbl 1238.11101); Abh. Math. Semin. Univ. Hamb. 80, No. 1, 47–57 (2010; Zbl 1214.11125); Proc. Japan Acad., Ser. A 85, No. 10, 155–159 (2009; Zbl 1234.11146)], in particular in [Zbl 1238.11101] they proved: Let \(p\) be a prime number, \(l\) a prime number different from \(p\), \(f\) the inertia degree of \(l\) in \(\mathbb Q(\mu_{2p}/\mathbb Q)\) and \(p^s\) the exact power of \(p\) dividing \(l^f-1\). Then there exists an explicit positive constant \(H(p,s,f)\) such that \(l\) does not divide \(h_{p,n}\) for any positive integer \(n\) if \(l\) does not divide \(h_{p,s-1}\) and is greater than \(H(p,s,f)\).
In this paper the authors improve the bound of this result of K. Horie and M. Horie for the prime number \(l\) for any odd prime \(p\).
Theorem 1. Let \(p\) be an odd prime number, \(l\) a prime different from \(p\) and \(n\) a positive integer. Choose \(s\) so that \(p^s\) is the exact power of \(p\) dividing \(l^{p-1}-1\). Let \(r:=\min\{n,s\}\) and \(c:=(p-1)\cdot p^{r-1}\). Let \(f\) be the inertia degree of \(l\) in \(\mathbb Q(\mu_p)/\mathbb Q\) and \[ G_1(p,r,f):=\Big(\Big(\frac{\sqrt{6}p}{2}\Big)^c\cdot c!\Big)^{1/f}. \] If \(l\) satisfies \(l>G_1(p,r,f)\), then \(l\) does not divide \(h_{p,n}\).
A further improvement as follows:
Let \(p,l,n,s,r,c\) and \(f\) be the same as in Theorem 1. Let \[ G_{cyclo}(p,r,f):=\Big(\sqrt{6}^c\Big(\frac{p^{p-2}((p-1)/2)!^2}{(p-1)!}\Big)^{c/(p-1)}c!\Big)^{1/f}. \] If \(l\) satisfies \(G_{cyclo}(p,r,f)\), then \(l\) does not divide \(h_{p,n}.\)
The proofs involve \(\mathbb Z_p\)-extensions theory, Horie units, Mahler measure, Minkowski geometry of numbers.

MSC:

11R29 Class numbers, class groups, discriminants
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R18 Cyclotomic extensions
11H06 Lattices and convex bodies (number-theoretic aspects)
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References:

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