Simultaneous indivisibility of class numbers of pairs of real quadratic\n fields

2020/12/02 by Jaitra Chattopadhyay, Chattopadhyay, Jaitra, Anupam Saikia +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.01202

openalex publication_date 2020/12/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

For a square-free integer t, Byeon citebyeon proved the existence of\ninfinitely many pairs of quadratic fields \ℚ(\√(D)) and\n\ℚ(\√(tD)) with D > 0 such that the class numbers of all of them\nare indivisible by 3. In the same spirit, we prove that for a given integer\nt \≥ 1 with t \≡ 0 pmod 4, a positive proportion of fundamental\ndiscriminants D > 0 exist for which the class numbers of both the real\nquadratic fields \ℚ(\√(D)) and \ℚ(\√(D + t)) are\nindivisible by 3. This also addresses the complement of a weak form of a\nconjecture of Iizuka in citeiizuka. As an application of our main result, we\nobtain that for any integer t \≥ 1 with t \≡ 0 pmod12, there are\ninfinitely many pairs of real quadratic fields \ℚ(\√(D)) and\n\ℚ(\√(D + t)) such that the Iwasawa \λ-invariants\nassociated with the basic \ℤ₃-extensions of both\n\ℚ(\√(D)) and \ℚ(\√(D + t)) are 0. For p = 3,\nthis supports Greenberg's conjecture which asserts that \λ_p(K) = 0\nfor any prime number p and any totally real number field K.\n

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• On the 3-rank of the class group of quadratic fields

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