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Some congruences connecting quadratic class numbers with continued fractions. (English) Zbl 1504.11120

Summary: Let \(p\) be a prime number, and \(h(-p)\) and \(h(p)\) be the ideal class numbers of the quadratic fields \(\mathbb{Q}(\sqrt{-p})\) and \(\mathbb{Q}(\sqrt{p})\) respectively. We prove that if \(p\equiv 1 \pmod 8\) then \(h(-p)\equiv h(p)m(4p) \pmod 8\), and if \(p\equiv 5 \pmod 8\) then \(h(-p)\equiv h(p)m(4p) \pmod 4\) under some further restrictions on the fundamental unit of \(\mathbb{Q}(\sqrt{p})\), where \(m(4p)\) is an integer depending on the minimal period of the negative continued fraction expansion of \(\sqrt{4p}\).

MSC:

11R29 Class numbers, class groups, discriminants
11A55 Continued fractions
11F20 Dedekind eta function, Dedekind sums
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[1] E. Brown, The power of 2 dividing the class-number of a binary quadratic discriminant, J. Number Theory 5 (1973), 413-419. · Zbl 0273.12005
[2] E. Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107. · Zbl 0287.12004
[3] L. Chua, B. Gunby, S. Park, and A. Yuan, Proof of a conjecture of Guy on class numbers, Int. J. Number Theory 11 (2015), 1345-1355. · Zbl 1325.11120
[4] H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993. · Zbl 0786.11071
[5] H. Cohn and G. Cooke, Parametric form of an eight class field, Acta Arith. 30 (1976), 367-377. · Zbl 0299.12004
[6] A. Eustis, The negs and regs of continued fractions, senior thesis, Harvey Mudd College, 2006; https://www.math.hmc.edu/seniorthesis/archives/2006/aeustis/aeustis2006-thesis.pdf.
[7] K. Feng, Algebraic Number Theory, Science Press, Beijing, 2000 (in Chinese).
[8] A. Fröhlich and M. J. Taylor, Algebraic Number Theory, Cambridge Stud. Adv. Math. 27, Cambridge Univ. Press, Cambridge, 1993.
[9] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., edited and revised by D. R. Heath-Brown and J. H. Silverman, Oxford Univ. Press, Oxford, 2008. · Zbl 1159.11001
[10] F. Herzog, On the continued fractions of conjugate quadratic irrationalities, Canad. Math. Bull. 23 (1980), 199-206. · Zbl 0432.10004
[11] F. E. P. Hirzebruch, Hilbert modular surfaces, Enseign. Math. (2) 19 (1973), 183-281. · Zbl 0285.14007
[12] L.-K. Hua, Introduction to Number Theory, Springer, Berlin, 1982. · Zbl 0483.10001
[13] H. Lu, Kronecker limit formula of real quadratic fields. I, Sci. Sinica Ser. A 27 (1984), 1233-1250. · Zbl 0564.12005
[14] H. Lu, Hirzebruch sum and class number of the quadratic fields, Chinese Sci. Bull. 36 (1991), 1145-1147. · Zbl 0764.11044
[15] H. Lu, Gauss’s Conjectures on Quadratic Number Fields, Shanghai Scientific and Technical Publ., Shanghai, 1994 (in Chinese). · Zbl 1107.11300
[16] H. Lu, C. Ji, and R. Jiao, Kronecker limit formula for real quadratic number fields. III, Sci. China Ser. A 44 (2001), 1132-1138. · Zbl 1044.11098
[17] H. Lu and M. Zhang, On Kronecker limit formula for real quadratic fields. II, Sci. China Ser. A 32 (1989), 1409-1422. · Zbl 0713.11079
[18] C. Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957. · Zbl 0079.06001
[19] O. Perron, Die Lehre von den Kettenbrüchen. Bd. I. Elementare Kettenbrüche, 3th ed., B. G. Teubner, Stuttgart, 1954. · Zbl 0056.05901
[20] H. Rademacher and E. Grosswald, Dedekind Sums, Carus Math. Monogr. 16, Math. Assoc. America, Washington, DC, 1972. · Zbl 0251.10020
[21] J. J. Rotman, A First Course in Abstract Algebra, Prentice-Hall, Upper Saddle River, NJ, 1996. · Zbl 0847.00004
[22] C. L. Siegel, Advanced Analytic Number Theory, 2nd ed., Tata Inst. Fund. Res. Stud. Math. 9, Tata Institute of Fundamental Research, Bombay, 1980.√ · Zbl 0478.10001
[23] K. S. Williams, The class number of Q(p) modulo 4, for p ≡ 5 (mod 8) a prime, Pacific J. Math. 92 (1981), 241-248. · Zbl 0408.12008
[24] D. Zagier, A Kronecker limit formula for real quadratic fields, Math. Ann. 213 (1975), 153-184. · Zbl 0283.12004
[25] D. Zagier, Nombres de classes et fractions continues, Astérisque 24-25 (1975), 81-97. · Zbl 0309.12002
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