×

Sum of three squares and class numbers of imaginary quadratic fields. (English) Zbl 1256.11058

The author proves that, for given positive integers \(k\) and suitably chosen integers \(a\) and \(M\), there exist infinitely many positive squarefree integers \(d\) such that \(h(-d)\) is divisible by \(k\) with \(d \equiv a \bmod M\). Using a result of Gauss, this divisibility result is then transferred to the number of representations of integers as sums of three squares.

MSC:

11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] N. C. Ankeny and S. Chowla, On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955), 321-324. · Zbl 0065.02402
[2] W. Kohnen and K. Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math. 135 (1999), no. 2, 387-398. · Zbl 0931.11044
[3] S.-N. Kuroda, On the class number of imaginary quadratic number fields, Proc. Japan Acad. 40 (1964), 365-367. · Zbl 0128.03404
[4] T. Nagel, Über die Klassenzahl imäginar-quadratischer Zahkörper, Abh. Math. Seminar Univ. Hamburg 1 (1922) 140-150. · JFM 48.0170.03
[5] O. Perron, Bemerkungen über die Verteilung der quadratischen Reste, Math. Z. 56 (1952), 122-130. · Zbl 0048.03002
[6] K. Soundararajan, Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc. (2) 61 (2000), no. 3, 681-690. · Zbl 1018.11054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.