Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields


Authors: Byeong Moon Kim, Ji Young Kim and Poo-Sung Park
Journal: Math. Comp. 79 (2010), 1123-1144
MSC (2000): Primary 11E39; Secondary 11E20, 11E41
DOI: https://doi.org/10.1090/S0025-5718-09-02287-X
Published electronically: July 16, 2009
MathSciNet review: 2600559
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields $ \mathbb{Q}(\sqrt{-m})$ for all $ m$. For each imaginary quadratic field $ \mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice $ L$ represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then $ L$ is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13. In addition, we determine the minimal rank of universal Hermitian lattices for all imaginary quadratic fields.


References [Enhancements On Off] (What's this?)

  • 1. M. Bhargava, On the Conway-Schneeberger fifteen theorem, Contemp. Math. 272 (2000), 27-37. MR 1803359 (2001m:11050)
  • 2. H. Brandt, O. Intrau, Tabellen reduzierter positiver ternärer quadratischer Formen, Akademie-Verlag, Berlin, 1958. MR 0106204 (21:4938)
  • 3. W. K. Chan, M.-H. Kim, S. Raghavan, Ternary universal integral quadratic forms over real quadratic fields, Japan. J. Math. 22 (1996), 263-273. MR 1432376 (97m:11051)
  • 4. J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemp. Math. 272 (2000), 23-26. MR 1803358 (2001m:11049)
  • 5. L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. MR 1561323
  • 6. A. G. Earnest, A. Khosravani, Universal binary Hermitian forms, Math. Comp. 66 (1997), 1161-1168. MR 1422787 (98a:11048)
  • 7. F. Götzky, Über eine Zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlicher, Math. Ann. 100 (1928), 411-437. MR 1512493
  • 8. J. Hanke, Some recent results about (ternary) quadratic forms, Number theory, CRM Proc. Lecture Notes, 36 (2004), 147-164. MR 2076591 (2005k:11076)
  • 9. J. Hanke, http://www.math.duke.edu/~jonhanke/290/Universal-290.html.
  • 10. H. Iwabuchi, Universal binary positive definite Hermitian lattices, Rocky Mountain J. Math. 30 (2000), no.3, 951-959. MR 1797825 (2002c:11043)
  • 11. N. Jacobson, A note on hermitian forms, Bull. Amer. Math. Soc. 46 (1940), 264-268. MR 0001957 (1:325d)
  • 12. B. M. Kim, Positive universal forms over totally real fields, Ph.D. Thesis, Seoul National Univ., 1997.
  • 13. B. M. Kim, Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms, Manuscripta Math. 99 (1999), 181-184. MR 1697212 (2000h:11032)
  • 14. B. M. Kim, Universal octonary diagonal forms over some real quadratic fields, Comment. Math. Helv. 75 (2000), 410-414. MR 1793795 (2001m:11046)
  • 15. B. M. Kim, M.-H. Kim, B.-K. Oh, $ 2$-universal positive definite integral quinary quadratic forms, Contemp. Math. 249 (1999), 51-62. MR 1732349 (2001c:11047)
  • 16. B. M. Kim, M.-H. Kim, B.-K. Oh, A finiteness theorem for representability of quadratic forms by forms, J. Reine Angew. Math. 581 (2005), 23-30. MR 2132670 (2006a:11044)
  • 17. J.-H. Kim, P.-S. Park, A few uncaught universal Hermitian forms, Proceedings of Amer. Math. Soc. 135 (2007), 47-49. MR 2280173 (2007i:11055)
  • 18. M.-H. Kim, Recent developments on universal forms, Contemp. Math. 344 (2004), 215-228. MR 2058677 (2005c:11042)
  • 19. J. L. Lagrange, Démonstration d'un théorème d'arithmétique, Œuvres 3 (1770), 189-201.
  • 20. H. Maass, Über die Darstellung total positiver Zahlen des Körpers $ \mathbf{R}(\sqrt{5})$ als Summe von drei Quadraten, Abh. Math. Sem. Hamburg 14 (1941), 185-191. MR 0005505 (3:163a)
  • 21. O. T. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1973.
  • 22. P.-S. Park, $ 2$-Universal Hermitian Forms, Ph.D. Thesis, Seoul National Univ., 2005.
  • 23. S. Ramanujan, On the expression of a number in the form $ ax^2+by^2+cz^2+dw^2$, Proc. Cambridge Phil. Soc. 19 (1917), 11-21.
  • 24. A. Rokicki, Finiteness results for definite $ n$-regular and almost $ n$-regular hermitian forms, Ph.D. Thesis, Wesleyan Univ., 2005.
  • 25. W. A. Schneeberger, Arithmetic and Geometry of Integral Lattices, Ph.D. Thesis, Princeton Univ., 1995.
  • 26. M. F. Willerding, Determination of all classes of positive quaternary quadratic forms which represent all (positive) integers, Bull. Amer. Math. Soc. 54 (1948), 334-337. MR 0024939 (9:571e)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11E39, 11E20, 11E41

Retrieve articles in all journals with MSC (2000): 11E39, 11E20, 11E41


Additional Information

Byeong Moon Kim
Affiliation: Department of Mathematics, Kangnung-Wonju National University, 123 Chibyondong, Gangneung, Gangwon-Do, Korea
Email: kbm@nukw.ac.kr

Ji Young Kim
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea
Email: jykim@kias.re.kr

Poo-Sung Park
Affiliation: Department of Mathematics Education, Kyungnam University, Masan, Kyungnam, 631-701, Korea
Email: pspark@kyungnam.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-09-02287-X
Keywords: Universal Hermitian form
Received by editor(s): March 28, 2008
Received by editor(s) in revised form: April 14, 2009
Published electronically: July 16, 2009
Additional Notes: The first named author was supported by the Korean Council for University Education, grant funded by Korean Government (MOEHRD) for 2006 Domestic Faculty Exchange.
The second and the third named authors were partially supported by KRF(2005-070-c00004).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society