ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11R18, 11R29, 11B68 Kimura, Noriaki, College of Industrial Technology, Nihon University A simple Proof of Carlitz’s congruence of Vandiver Type Let p be an odd prime, Q the rational number field. And let ζ_{p} be the pth root of unity. Let p − 1 = ab, where b is odd > 1. Let K be the imaginary cyclic subfield of Q(ζ_{p}) of degree a. For any integer n ≥ 1, Carlitz has proved the following congruence of Vandiver Type.

h^{−}(K) ≡ (−1)^{\frac{a}{2}} 2^{− \frac{a−2}{2}} ∏_{{s=1 \atop s:odd}}^{a−1} B_{bsp^{n}+1} (mod p^{n}),

where B_{bsp^{n}+1} are Bernoulli numbers. The simplest proof of this congruence will be presented. Also the similar congruence will be considered in case K is an imaginary subfield of Q(ζ_{p^{ρ}}) (ρ ≥ 2). References: H. S. Vandiver, On the first factor of the class number of a cyclotomic field, Bull. Amer. Math. Soc.25, 1919, 458–461. L. Carlitz, The first factor of the class number of a cyclic field, Canad. J. Math.6, 1954, 23–26. I. Sh. Slavutsky, The simplest proof of Vandiver’s theorem, Acta Arith.15, 1969, 117–118.