An upper bound for the λ-invariant of imaginary abelian fields

Mathematische Annalen -     

Iwasawa invariants and Kummer conguruences

Let f(x, χ) be the Iwasawa power series for the p-adic L-function L_p(s, χ), where χ is an even nonprincipal character with conductor not divisible by p² (or by 8, when p = 2). The divisibility by p of the first p coefficients of f(x, χ) is characterized by Kummer type congruences of generalized Bernoulli numbers. Applications to Iwasawa invariants and class numbers of imaginary Abelian fields are discussed.     

A short proof for the nonvanishing of a character sum

Let χ be an odd residue class character with conductor f (a prime power) and order 2^h. It is shown that the nonvanishing of the sum χ(1) + 2χ(2) + … + fχ(f) follows directly from properties of the minimum polynomial of a 2^hth root of unity.