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RELATIVE CLASS NUMBERS INSIDE THE pTH CYCLOTOMIC FIELD

Ichimura, Humio 大阪大学 DOI:10.18910/77238

2020.10

概要

For a prime number p ≡ 3 mod 4, we write p = 2nℓ^f + 1 for some power ℓ^f of an odd prime number ℓ and an odd integer n with ℓ∤ n. For 0 ≤ t ≤ f , let K_t be the imaginary subfield of ℚ(ζp) of degree 2ℓ^t and let h−_t be the relative class number of K_t. We show that for n = 1 (resp.n ≥ 3), a prime number r does not divide the ratio h−_t / h−_＜t−1＞ when r is a primitive root modulo ℓ^2 and r ≥ ℓ＜f−t＞ − 1 (resp. r ≥ (n − 2)ℓ＜f−t＞ + 1). In particular, for n = 1 or 3, the ratio h−_f / h−＜f−1＞ at the top is not divisible by r whenever r is a primitive root modulo ℓ^2. Further, we show that the ℓ-part of h−_t / h−_＜t−1＞ stabilizes for “large” t under some assumption.

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