On Small Iwasawa Invariants and Imaginary Quadratic Fields

On Small Iwasawa Invariants and Imaginary Quadratic Fields

If p is an odd prime that does not divide the class number of the imaginary quadratic field k , and the cyclotomic Z -extension of k has A-invariant less than or equal to two, we prove that every totally ramified Z extension of k has //-invariant equal to zero and A-invariant less than or equal to two. Combined with a result of Bloom and Gerth, this has the consequence that ß = 0 for every Z -e...

Visualizing imaginary quadratic fields

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...

The Iwasawa Invariants and the Higher K-Groups Associated to Real Quadratic Fields

First, let us look back over the case of χ = χ the trivial character. For k = 1, Bk,χ0 is the kth Bernoulli number Bk, and B1,χ0 = B1 + 1 = 1/2. A pair of integers (p, k) is said to be an irregular pair if p is a prime, k is an even integer satisfying 2 ≤ k ≤ p−3, and p divides the numerator of Bk = Bk,χ0 . Irregular pairs have been computed by Kummer, Vandiver, D.H. Lehmer, E. Lehmer, Selfridg...

On Congruent Primes and Class Numbers of Imaginary Quadratic Fields

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a relat...

On the non-triviality of the basic Iwasawa λ-invariant for an infinitude of imaginary quadratic fields