TY - JOUR

T1 - On the Iwasawa λ-invariant of the cyclotomic Z2-extension of Q(√p), III

AU - Fukuda, Takashi

AU - Komatsu, Keiichi

AU - Ozaki, Manabu

AU - Tsuji, Takae

PY - 2016

Y1 - 2016

N2 - In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic Z2-extension of k = Q(√p ) with prime number p. Criteria and numerical algorithm in [5], [3] and [6] enable us to show λ2(k) = 0 for all p less than 105 except p =13841; 67073. All the known criteria at present can not handle p = 13841; 67073. In this paper, we develop another criterion for λ2(k) = 0 using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that λ2(Q(√p )) = 0 for p = 13841; 67073. So we announce here that λ2(Q(√p)) = 0 for all prime numbers p less that 105.

AB - In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic Z2-extension of k = Q(√p ) with prime number p. Criteria and numerical algorithm in [5], [3] and [6] enable us to show λ2(k) = 0 for all p less than 105 except p =13841; 67073. All the known criteria at present can not handle p = 13841; 67073. In this paper, we develop another criterion for λ2(k) = 0 using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that λ2(Q(√p )) = 0 for p = 13841; 67073. So we announce here that λ2(Q(√p)) = 0 for all prime numbers p less that 105.

KW - Cyclotomic unit

KW - Iwasawa invariant

KW - Real quadratic field

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U2 - 10.7169/facm/2016.54.1.1

DO - 10.7169/facm/2016.54.1.1

M3 - Article

AN - SCOPUS:84962124272

VL - 54

SP - 7

EP - 17

JO - Functiones et Approximatio, Commentarii Mathematici

JF - Functiones et Approximatio, Commentarii Mathematici

SN - 0208-6573

IS - 1

ER -