A note on basic Iwasawa λ-invariants of imaginary quadratic fields and congruence of modular forms

Abstract

A note on basic Iwasawa λ-invariants of imaginary quadratic fields and congruence of modular forms by Dongho Byeon (Seoul)1. Introduction and statement of results. For a number field k and a prime number l, we denote by h(k) the class number of k and by λ l (k) the Iwasawa λ-invariant of the cyclotomic Z l -extension of k, where Z l is the ring of l-adic integers.Let l be an odd prime number. Using the Kronecker class number relation for quadratic forms, Hartung [3] proved that there exist infinitely many imaginary quadratic fields k whose class numbers are not divisible by l. For the case l = 2, this is an immediate consequence of Gauss' genus theory. On the other hand, using the idea of Hartung and Eichler's trace formula combined with the l-adic Galois representation attached to the Jacobian variety J = J 0 (l) of the modular curve X = X 0 (l), Horie [4] proved that there exist infinitely many imaginary quadratic fields k such that l does not split in k and l does not divide h(k). Later Horie and Onishi [5] obtained more refined results. By a theorem of Iwasawa [6], these results imply that there exist infinitely many imaginary quadratic fields k with λ l (k) = 0. For the case l = 2, this is also an immediate consequence of Gauss' genus theory. For the case l = 3, by refining Davenport and Heilbronn's result [2], Nakagawa and Horie [8] gave a positive lower bound on the density of imaginary quadratic fields k and real quadratic fields k with λ l (k) = 0.

A Note on the Existence of Certain Infinite Families of Imaginary Quadratic Fields

“…In this note, as the author's previous work [2], refining Kohnen and Ono's method [3,5] which use Sturm's result [6] on the congruence of modular forms, we will give another proof of the above theorem and go a step further by obtaining the following estimate. …”

On certain infinite families of imaginary quadratic fields whose Iwasawa λ-invariant is equal to 1

“…The idea of the proof is used widely in the study of indivisibility of the class number of quadratic fields (cf. W. Kohnen and K. Ono [21], K. Ono [25], D. Byeon [2,3,4], I. Kimura [19], etc). First, we sketch the outline of the proof.…”

“…Both d 1 s 2 0 + q 2 5 = p t 0 and 3d 1 s 2 0 − q 2 5 = ±1 hold for some positive integers s 0 and t 0 . Then, we have 4q 2 5 = 3p t 0 ± 1. This is a contradiction with Lemma 4.6.…”

“…Therefore, we treat the case where q 6 = 3. First, we treat the equation 16q 2 6 − 3p x = −1. Since q 2 6 ≡ 1 mod 3 holds, we have…”

“…Here, we give some additional explanation for the case where x 1 = 2. Note that 2 p−1 ≡ −1 mod p 2 . In fact, if 2 p−1 ≡ −1 mod p 2 , the congruence relation 2 p−1 + 1 ≡ 0 mod p and 2 p−1 − 1 ≡ 0 mod p hold.…”