Computation of Iwasawa Lambda invariants for imaginary quadratic fields

Dummit, David S.; Ford, David; Kisilevsky, Hershy; Sands, Jonathan W.

doi:10.1016/s0022-314x(05)80027-7

Cited by 19 publications

(14 citation statements)

References 10 publications

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“…Table 9. 10 9 < d < 10 9 + 10 6 , d ≡ 0 (mod 3) In the other cases λ ≥ 1, and we used the formulas from [2]. In the cases considered in the present paper, the formulas are as follows.…”

Section: Data On λ Invariantsmentioning

confidence: 99%

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Heuristics for class numbers and lambda invariants

Kraft

Washington

2006

Math. Comp.

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Abstract. Let K = Q( √ −d) be an imaginary quadratic field and let Q( √ 3d) be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz's theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as d varies. We deduce heuristic predictions for the behavior of the Iwasawa λ-invariant for the cyclotomic Z 3 -extension of K and test them computationally.The Cohen-Lenstra heuristics [1] give predictions for frequencies of class numbers and class groups of number fields. In the following, we investigate a related situation and a more specific question:I. Are there heuristics for the Iwasawa lambda invariants, similar to those of Cohen and Lenstra for class groups of number fields? The λ 2 -invariants of imaginary quadratic fields are given by a simple formula of Ferrero [4] and Kida [5] and are correspondingly not suitable for a heuristic analysis. We therefore consider the first nontrivial case, namely the λ-invariant for the cyclotomic Z 3 -extension of an imaginary quadratic field K as K varies. When 3 does not split in K, the frequency of λ = 0 is easy to treat. We give a prediction for the frequency of λ = 1 in this case. We also compute numerical data that agrees fairly well with the prediction; however, it is well known that the convergence of empirical data to the CohenLenstra heuristics is quite slow, and we presumably have a similar slowness in the present situation. Therefore, any numerical agreement or disagreement cannot necessarily be regarded as decisive. We also collect data for the case when 3 splits in the imaginary quadratic field. In this case, we always have λ ≥ 1. It appears that the frequencies of a given λ are similar to those for λ − 1 in the nonsplit case. We regard this as pointing towards some type of theoretical model for λ heuristics, similar to the idea of weighting by the inverse of the size of automorphism groups in the Cohen-Lenstra setting.II. It was proved in [6] that if 3 splits in an imaginary quadratic field Q( √ −d) and if 3 divides the class number of Q( √ 3d), then λ ≥ 2. Since it might be suspected that λ tends to be small, this could indicate that 3 divides the class number of Q( √ 3d), with d ≡ 2 (mod 3), with less than the frequency predicted by CohenLenstra heuristics for class numbers of all real quadratic fields. Nevertheless, our numerical experiments do not indicate the presence of any such bias.In Section 9, we give some data for the distribution of λ 3 -invariants of imaginary quadratic fields Q( √ −d). As our analysis shows, it is natural to break into cases

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Section: Data On λ Invariantsmentioning

confidence: 99%

“…Note that the formula in [2] is missing the term (1 − l 2 ) 3 /9 in the binomial coefficient. This corresponds to the fact that a term (jp) 3 /3 is missing in the calculation of log p (i) in [2, page 104].…”

Section: Data On λ Invariantsmentioning

confidence: 99%

Heuristics for class numbers and lambda invariants

Kraft

Washington

2006

Math. Comp.

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“…Let I < p -1. The following are equivalent: The computational results of [4] provide many examples where condition (b) of this proposition holds. In fact, we usually find that Xc < 2, and conclude from the proposition that \A(K¡)\ < \A(k)\p2.…”

Section: Overviewmentioning

confidence: 97%

On Small Iwasawa Invariants and Imaginary Quadratic Fields

Sands¹

1991

Proceedings of the American Mathematical Society

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Abstract.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 Zextension 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 -extension of k , under the same assumptions. In the principal case under consideration, Iwasawa's formula for the power of p in the class number of the «th layer of a Z -extension becomes valid for all n , and is completely explicit.

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“…It remains to consider the situation of D = 3 mod 8 and p a split prime in k, which is more interesting. Let F denote the Hilbert class field of k and let F (2 ) denote the ray class field of k of conductor (2). Let G = Gal (F (2 )/ k) be the Galois group of FQ) over k (the ray class group to conductor (2)).…”

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confidence: 99%

“…Let F denote the Hilbert class field of k and let F (2 ) denote the ray class field of k of conductor (2). Let G = Gal (F (2 )/ k) be the Galois group of FQ) over k (the ray class group to conductor (2)). Since D = 3 mod 8 the ideal (2) is inert in k so that F(2> is an abelian extension of F of degree 3 with Galois group H = Gal (F (2 )/ F) canonically isomorphic by the Artin isomorphism to the group (0/ (2)0)* ~ Z/3Z, with representatives 1, UJ , 1 + UJ mod (2).…”

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confidence: 99%

The Parity Distribution of Traces in Imaginary Quadratic Fields

Dummit¹

1991

Can. math. bull.

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Computations of the Iwasawa λ -invariant for imaginary quadratic fields showed a discrepancy in the proportion of even and odd traces of certain integers from these imaginary quadratic fields. This paper shows that such a discrepancy is in some sense to be expected and that the proportion of even and odd traces of principal generators of powers of prime ideals in imaginary quadratic fields is related to the 3-primary component of the class group.

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Computation of Iwasawa Lambda invariants for imaginary quadratic fields

Cited by 19 publications

References 10 publications

Heuristics for class numbers and lambda invariants

Heuristics for class numbers and lambda invariants

On Small Iwasawa Invariants and Imaginary Quadratic Fields

The Parity Distribution of Traces in Imaginary Quadratic Fields

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