On primitively 2-universal quadratic forms
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N. V. Budarina
Translated by: N. B. Lebedinskaya - St. Petersburg Math. J. 23 (2012), 435-458
- DOI: https://doi.org/10.1090/S1061-0022-2012-01203-3
- Published electronically: March 2, 2012
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Abstract:
The primitive representations of binary positive definite, classically integral quadratic forms over the local rings $\mathbb Z_p$ are studied. For the global ring, an efficient method is obtained for determining when a quadratic form is primitively 2-universal.References
- Manjul Bhargava, On the Conway-Schneeberger fifteen theorem, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 27–37. MR 1803359, DOI 10.1090/conm/272/04395
- Byeong-Kweon Oh, Universal $\textbf {Z}$-lattices of minimal rank, Proc. Amer. Math. Soc. 128 (2000), no. 3, 683–689. MR 1654105, DOI 10.1090/S0002-9939-99-05254-5
- Byeong Moon Kim, Myung-Hwan Kim, and Byeong-Kweon Oh, $2$-universal positive definite integral quinary quadratic forms, Integral quadratic forms and lattices (Seoul, 1998) Contemp. Math., vol. 249, Amer. Math. Soc., Providence, RI, 1999, pp. 51–62. MR 1732349, DOI 10.1090/conm/249/03747
- Wai-kiu Chan, Myung-Hwan Kim, and S. Raghavan, Ternary universal integral quadratic forms over real quadratic fields, Japan. J. Math. (N.S.) 22 (1996), no. 2, 263–273. MR 1432376, DOI 10.4099/math1924.22.263
- A. G. Earnest and Azar Khosravani, Universal positive quaternary quadratic lattices over totally real number fields, Mathematika 44 (1997), no. 2, 342–347. MR 1600557, DOI 10.1112/S0025579300012651
- Byeong Moon Kim, Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms, Manuscripta Math. 99 (1999), no. 2, 181–184. MR 1697212, DOI 10.1007/s002290050168
- Byeong Moon Kim, Universal octonary diagonal forms over some real quadratic fields, Comment. Math. Helv. 75 (2000), no. 3, 410–414. MR 1793795, DOI 10.1007/s000140050133
- Hans Maass, Über die Darstellung total positiver Zahlen des Körpers $R(5)$ als Summe von drei Quadraten, Abh. Math. Sem. Hansischen Univ. 14 (1941), 185–191 (German). MR 5505, DOI 10.1007/BF02940744
- Byeong-Kweon Oh, The representation of quadratic forms by almost universal forms of higher rank, Math. Z. 244 (2003), no. 2, 399–413. MR 1992544, DOI 10.1007/s00209-003-0505-3
- H. D. Kloosterman, On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$, Acta Math. 49 (1926), 407–464.
- Gordon Pall, The completion of a problem of Kloosterman, Amer. J. Math. 68 (1946), 47–58. MR 14377, DOI 10.2307/2371739
- Arnold E. Ross and Gordon Pall, An extension of a problem of Kloosterman, Amer. J. Math. 68 (1946), 59–65. MR 14378, DOI 10.2307/2371740
- J. Bochnak and B.-K. Oh, Almost-universal quadratic forms: an effective solution of a problem of Ramanujan, Duke Math. J. 147 (2009), no. 1, 131–156. MR 2494458, DOI 10.1215/00127094-2009-008
- M. Bhargava, Finiteness theorems for quadratic forms, Preprint.
- Byeong Moon Kim, Myung-Hwan Kim, and Byeong-Kweon Oh, A finiteness theorem for representability of quadratic forms by forms, J. Reine Angew. Math. 581 (2005), 23–30. MR 2132670, DOI 10.1515/crll.2005.2005.581.23
- Byeong-Kweon Oh, Positive definite $n$-regular quadratic forms, Invent. Math. 170 (2007), no. 2, 421–453. MR 2342641, DOI 10.1007/s00222-007-0068-8
- A. G. Earnest, The representation of binary quadratic forms by positive definite quaternary quadratic forms, Trans. Amer. Math. Soc. 345 (1994), no. 2, 853–863. MR 1264145, DOI 10.1090/S0002-9947-1994-1264145-0
- N. Budarina, On primitively universal quadratic forms, Lith. Math. J. 50 (2010), no. 2, 140–163. MR 2653643, DOI 10.1007/s10986-010-9076-2
- V. G. Zhuravlev, Representation of a form by a genus of quadratic forms, Algebra i Analiz 8 (1996), no. 1, 21–112 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 1, 15–84. MR 1392012
- V. G. Zhuravlëv, Orbits of representations of numbers by local quadratic forms, Tr. Mat. Inst. Steklova 218 (1997), no. Anal. Teor. Chisel i Prilozh., 151–164 (Russian); English transl., Proc. Steklov Inst. Math. 3(218) (1997), 146–159. MR 1642373
- V. G. Zhuravlëv, Embedding of $p$-elementary lattices, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 1, 77–106 (Russian, with Russian summary); English transl., Izv. Math. 63 (1999), no. 1, 73–102. MR 1701839, DOI 10.1070/im1999v063n01ABEH000229
- V. G. Zhuravlev, Primitive embeddings into local lattices of prime determinant, Algebra i Analiz 11 (1999), no. 1, 87–117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 1, 67–90. MR 1691081
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
- J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
- V. G. Zhuravlev, Deformations of quadratic Diophantine systems, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 6, 15–56 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 6, 1085–1126. MR 1892903, DOI 10.1070/IM2001v065n06ABEH000364
Bibliographic Information
- N. V. Budarina
- Affiliation: Khabarovsk Division, Institute of Applied Mathematics, Russian Academy of Sciences, 54 Dzerzhinsky Street, Khabarovsk 680000, Russia
- Email: buda77@mail.ru
- Received by editor(s): February 5, 2010
- Published electronically: March 2, 2012
- Additional Notes: Supported by RFBR (grants nos. 08-01-00326 and 07-01-00306)
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 435-458
- MSC (2010): Primary 11E08
- DOI: https://doi.org/10.1090/S1061-0022-2012-01203-3
- MathSciNet review: 2896164