Lower bound for class numbers of certain real quadratic fields

Mohit Mishra

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 1-14
  • ISSN: 0011-4642

Abstract

top
Let d be a square-free positive integer and h ( d ) be the class number of the real quadratic field ℚ ( 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 2 g + 1 is a square-free integer, then g ∣ 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 2 g + 1 ) = g . We also obtain a similar result for the family ℚ ( n 2 g + 4 ) . As another application, we deduce some criteria for a class group of prime power order to be cyclic.

How to cite

top

Mishra, Mohit. "Lower bound for class numbers of certain real quadratic fields." Czechoslovak Mathematical Journal 73.1 (2023): 1-14. <http://eudml.org/doc/299348>.

@article{Mishra2023,
abstract = {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.},
author = {Mishra, Mohit},
journal = {Czechoslovak Mathematical Journal},
keywords = {real quadratic field; class group; class number; Dedekind zeta values},
language = {eng},
number = {1},
pages = {1-14},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lower bound for class numbers of certain real quadratic fields},
url = {http://eudml.org/doc/299348},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Mishra, Mohit
TI - Lower bound for class numbers of certain real quadratic fields
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 1
EP - 14
AB - 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.
LA - eng
KW - real quadratic field; class group; class number; Dedekind zeta values
UR - http://eudml.org/doc/299348
ER -

References

top
  1. Ankeny, N. C., Chowla, S., 10.2140/pjm.1955.5.321, Pac. J. Math. 5 (1955), 321-324. (1955) Zbl0065.02402MR0085301DOI10.2140/pjm.1955.5.321
  2. Apostol, T. M., 10.1215/S0012-7094-50-01716-9, Duke Math. J. 17 (1950), 147-157. (1950) Zbl0039.03801MR0034781DOI10.1215/S0012-7094-50-01716-9
  3. Biró, A., 10.4064/aa107-2-5, Acta Arith. 107 (2003), 179-194. (2003) Zbl1154.11339MR1970822DOI10.4064/aa107-2-5
  4. Biró, A., 10.4064/aa106-1-6, Acta Arith. 106 (2003), 85-104. (2003) Zbl1154.11338MR1956977DOI10.4064/aa106-1-6
  5. Biró, A., Lapkova, K., 10.4064/aa118-2-2, Acta Arith. 172 (2016), 117-131. (2016) Zbl1358.11119MR3455622DOI10.4064/aa118-2-2
  6. Byeon, D., Kim, H. K., 10.1006/jnth.1996.0052, J. Number Theory 57 (1996), 328-339. (1996) Zbl0846.11060MR1382755DOI10.1006/jnth.1996.0052
  7. Byeon, D., Kim, H. K., 10.1006/jnth.1997.2059, J. Number Theory 62 (1997), 257-272. (1997) Zbl0871.11076MR1432773DOI10.1006/jnth.1997.2059
  8. Chakraborty, K., Hoque, A., Mishra, M., 10.2206/kyushujm.74.201, Kyushu J. Math. 74 (2020), 201-210. (2020) Zbl1452.11132MR4129806DOI10.2206/kyushujm.74.201
  9. Chakraborty, K., Hoque, A., Mishra, M., 10.1007/s40316-020-00139-1, Ann. Math. Qué. 45 (2021), 203-212. (2021) Zbl1469.11432MR4229182DOI10.1007/s40316-020-00139-1
  10. Chowla, S., Friedlander, J., 10.1017/S0017089500002718, Glasg. Math. J. 17 (1976), 47-52. (1976) Zbl0323.12006MR0417117DOI10.1017/S0017089500002718
  11. Hasse, H., 10.5169/seals-23925, Elem. Math. 20 (1965), 49-59 German. (1965) Zbl0128.03502MR0191889DOI10.5169/seals-23925
  12. Kim, H. K., Leu, M.-G., Ono, T., 10.3792/pjaa.63.222, Proc. Japan Acad., Ser. A 63 (1987), 222-224. (1987) Zbl0624.12002MR0907000DOI10.3792/pjaa.63.222
  13. Lang, H., 10.1515/crll.1968.233.123, J. Reine Angew. Math. 233 (1968), 123-175 German. (1968) Zbl0165.36504MR0238804DOI10.1515/crll.1968.233.123
  14. Lang, S., 10.1007/978-1-4612-0853-2, Graduate Texts in Mathematics 110. Springer, New York (1994). (1994) Zbl0811.11001MR1282723DOI10.1007/978-1-4612-0853-2
  15. Lapkova, K., 10.4064/aa153-3-4, Acta Arith. 153 (2012), 281-298. (2012) Zbl1329.11118MR2912719DOI10.4064/aa153-3-4
  16. Lemmermeyer, F., Algebraic Number Theory, Bilkent University, Bilkent (2006), Available at . (2006) 
  17. Mollin, R. A., 10.1090/S0002-9939-1986-0826478-X, Proc. Am. Math. Soc. 96 (1986), 545-550. (1986) Zbl0591.12007MR0826478DOI10.1090/S0002-9939-1986-0826478-X
  18. Mollin, R. A., 10.1090/S0002-9939-1987-0908645-0, Proc. Am. Math. Soc. 101 (1987), 439-444. (1987) Zbl0632.12006MR0908645DOI10.1090/S0002-9939-1987-0908645-0
  19. Mollin, R. A., 10.1017/S0027763000000738, Nagoya Math. J. 105 (1987), 39-47. (1987) Zbl0591.12005MR0881007DOI10.1017/S0027763000000738
  20. Mollin, R. A., Williams, H. C., 10.1090/S0002-9939-1988-0934844-9, Proc. Am. Math. Soc. 102 (1988), 794-796. (1988) Zbl0649.12005MR0934844DOI10.1090/S0002-9939-1988-0934844-9
  21. Siegel, C. L., Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 10 (1969), 87-102 German. (1969) Zbl0186.08804MR0252349
  22. Weinberger, P. J., 10.1016/0022-314X(73)90049-8, J. Number Theory 5 (1973), 237-241. (1973) Zbl0287.12007MR0335471DOI10.1016/0022-314X(73)90049-8
  23. Yokoi, H., 10.1017/S0027763000012939, Nagoya Math. J. 33 (1968), 139-152. (1968) Zbl0167.04401MR0233803DOI10.1017/S0027763000012939
  24. Yokoi, H., 10.1016/0022-314X(70)90010-7, J. Number Theory 2 (1970), 106-115. (1970) Zbl0201.05703MR0252351DOI10.1016/0022-314X(70)90010-7
  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. (1986) Zbl0612.12010MR0891892

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.