Article
| Title: | Lower bound for class numbers of certain real quadratic fields (English) |
| Author: | Mishra, Mohit |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 1 |
| Year: | 2023 |
| Pages: | 1-14 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $d$ be a square-free positive integer and $h(d)$ be the class number of the real quadratic field $\mathbb {Q}{(\sqrt {d})}.$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g \mid h(d)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb {Q}(\sqrt {n^{2g}+4})$. As another application, we deduce some criteria for a class group of prime power order to be cyclic. (English) |
| Keyword: | real quadratic field |
| Keyword: | class group |
| Keyword: | class number |
| Keyword: | Dedekind zeta values |
| MSC: | 11R11 |
| MSC: | 11R29 |
| MSC: | 11R42 |
| idZBL: | Zbl 07655753 |
| idMR: | MR4541087 |
| DOI: | 10.21136/CMJ.2022.0264-21 |
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| Date available: | 2023-02-03T11:06:02Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151500 |
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| Reference: | [1] Ankeny, N. C., Chowla, S.: On the divisibility of the class numbers of quadratic fields.Pac. J. Math. 5 (1955), 321-324. Zbl 0065.02402, MR 0085301, 10.2140/pjm.1955.5.321 |
| Reference: | [2] Apostol, T. M.: Generalized Dedekind sums and transformation formulae of certain Lambert series.Duke Math. J. 17 (1950), 147-157. Zbl 0039.03801, MR 0034781, 10.1215/S0012-7094-50-01716-9 |
| Reference: | [3] Biró, A.: Chowla's conjecture.Acta Arith. 107 (2003), 179-194. Zbl 1154.11339, MR 1970822, 10.4064/aa107-2-5 |
| Reference: | [4] Biró, A.: Yokoi's conjecture.Acta Arith. 106 (2003), 85-104. Zbl 1154.11338, MR 1956977, 10.4064/aa106-1-6 |
| Reference: | [5] Biró, A., Lapkova, K.: The class number one problem for the real quadratic fields $\mathbb{Q}(\sqrt{(an)^2+4a})$.Acta Arith. 172 (2016), 117-131. Zbl 1358.11119, MR 3455622, 10.4064/aa118-2-2 |
| Reference: | [6] Byeon, D., Kim, H. K.: Class number 1 criteria for real quadratic fields of Richaud-Degert type.J. Number Theory 57 (1996), 328-339. Zbl 0846.11060, MR 1382755, 10.1006/jnth.1996.0052 |
| Reference: | [7] Byeon, D., Kim, H. K.: Class number 2 criteria for real quadratic fields of Richaud-Degert type.J. Number Theory 62 (1997), 257-272. Zbl 0871.11076, MR 1432773, 10.1006/jnth.1997.2059 |
| Reference: | [8] Chakraborty, K., Hoque, A., Mishra, M.: A note on certain real quadratic fields with class number up to three.Kyushu J. Math. 74 (2020), 201-210. Zbl 1452.11132, MR 4129806, 10.2206/kyushujm.74.201 |
| Reference: | [9] Chakraborty, K., Hoque, A., Mishra, M.: On the structure of order 4 class groups of $\mathbb{Q} (\sqrt{n^2+1})$.Ann. Math. Qué. 45 (2021), 203-212. Zbl 1469.11432, MR 4229182, 10.1007/s40316-020-00139-1 |
| Reference: | [10] Chowla, S., Friedlander, J.: Class numbers and quadratic residues.Glasg. Math. J. 17 (1976), 47-52. Zbl 0323.12006, MR 0417117, 10.1017/S0017089500002718 |
| Reference: | [11] Hasse, H.: Über mehrklassige, aber eingeschlechtige reell-quadratische Zahlkörper.Elem. Math. 20 (1965), 49-59 German. Zbl 0128.03502, MR 0191889, 10.5169/seals-23925 |
| Reference: | [12] Kim, H. K., Leu, M.-G., Ono, T.: On two conjectures on real quadratic fields.Proc. Japan Acad., Ser. A 63 (1987), 222-224. Zbl 0624.12002, MR 0907000, 10.3792/pjaa.63.222 |
| Reference: | [13] Lang, H.: Über eine Gattung elemetar-arithmetischer Klasseninvarianten reell-quadratischer Zahlkörper.J. Reine Angew. Math. 233 (1968), 123-175 German. Zbl 0165.36504, MR 0238804, 10.1515/crll.1968.233.123 |
| Reference: | [14] Lang, S.: Algebraic Number Theory.Graduate Texts in Mathematics 110. Springer, New York (1994). Zbl 0811.11001, MR 1282723, 10.1007/978-1-4612-0853-2 |
| Reference: | [15] Lapkova, K.: Class number one problem for real quadratic fields of a certain type.Acta Arith. 153 (2012), 281-298. Zbl 1329.11118, MR 2912719, 10.4064/aa153-3-4 |
| Reference: | [16] Lemmermeyer, F.: Algebraic Number Theory.Bilkent University, Bilkent (2006), Available at {\def\let \relax \brokenlink{http://www.fen.bilkent.edu.tr/ franz/ant06/ant.pdf}}\kern0pt. |
| Reference: | [17] Mollin, R. A.: Lower bounds for class numbers of real quadratic fields.Proc. Am. Math. Soc. 96 (1986), 545-550. Zbl 0591.12007, MR 0826478, 10.1090/S0002-9939-1986-0826478-X |
| Reference: | [18] Mollin, R. A.: Lower bounds for class numbers of real quadratic and biquadratic fields.Proc. Am. Math. Soc. 101 (1987), 439-444. Zbl 0632.12006, MR 0908645, 10.1090/S0002-9939-1987-0908645-0 |
| Reference: | [19] Mollin, R. A.: On the insolubility of a class of Diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type.Nagoya Math. J. 105 (1987), 39-47. Zbl 0591.12005, MR 0881007, 10.1017/S0027763000000738 |
| Reference: | [20] Mollin, R. A., Williams, H. C.: A conjecture of S. Chowla via the generalized Riemann hypothesis.Proc. Am. Math. Soc. 102 (1988), 794-796. Zbl 0649.12005, MR 0934844, 10.1090/S0002-9939-1988-0934844-9 |
| Reference: | [21] Siegel, C. L.: Berechnung von Zetafunktionen an ganzzahligen Stellen.Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 10 (1969), 87-102 German. Zbl 0186.08804, MR 0252349 |
| Reference: | [22] Weinberger, P. J.: Real quadratic fields with class numbers divisible by $n$.J. Number Theory 5 (1973), 237-241. Zbl 0287.12007, MR 0335471, 10.1016/0022-314X(73)90049-8 |
| Reference: | [23] Yokoi, H.: On real quadratic fields containing units with norm -1.Nagoya Math. J. 33 (1968), 139-152. Zbl 0167.04401, MR 0233803, 10.1017/S0027763000012939 |
| Reference: | [24] Yokoi, H.: On the fundamental unit of real quadratic fields with norm 1.J. Number Theory 2 (1970), 106-115. Zbl 0201.05703, MR 0252351, 10.1016/0022-314X(70)90010-7 |
| Reference: | [25] Yokoi, H.: Class-number one problem for certain kind of real quadratic fields.Class Numbers and Fundamental Units of Algebraic Number Fields Nagoya University, Nagoya (1986), 125-137. Zbl 0612.12010, MR 0891892 |
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