Universal -lattices of minimal rank

Author:
Byeong-Kweon Oh

Journal:
Proc. Amer. Math. Soc. **128** (2000), 683-689

MSC (1991):
Primary 11E12, 11H06

DOI:
https://doi.org/10.1090/S0002-9939-99-05254-5

Published electronically:
July 6, 1999

MathSciNet review:
1654105

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

**[1]**E. Bannai,*Positive definite unimodular lattices with trivial automorphism groups*, Ohio State Univ., Thesis, 1988.**[2]**J.H. Conway, N.J.A. Sloane,*Sphere packings, lattices and groups*, Springer-Verlag, 1988. MR**89a:11067****[3]**-,*Low dimensional lattices. I. Quadratic forms of small determinant*, Proc. Royal. Soc. Lond. A.**418**(1988), 17-41. MR**90a:11071****[4]**-,*Low dimensional lattices. V. Integral coordinates for integral lattices*, Proc. Royal. Soc. Lond. A**426**(1989), 211-232. MR**90m:11100****[5]**J.H. Conway, W. Schneeberger,*A -theorem for universal quadratic forms*, to appear.**[6]**B.M. Kim, M-H. Kim, S. Raghavan,*-universal positive definite integral quinary diagonal quadratic forms*, Ramanujan J.**1**(1997), 333-337. CMP**98:10****[7]**B.M. Kim, M-H. Kim, B-K. Oh,*-universal positive definite integral quinary quadratic forms*, Preprint.**[8]**M-H. Kim, B-K. Oh,*Representations of positive definite senary integral-quadratic forms by a sum of squares*, J. Number Theory**63**(1997), 89-100. MR**98a:11045****[9]**-,*A lower bound for the number of squares whose sum represents integral quadratic forms*, J. Korean Math. Soc.**33**(1996), 651-655. MR**97j:11017****[10]**C. Ko,*On the representation of a sum of squares of linear forms*, Quart. J. Math. Oxford**8**(1937), 81-98.**[11]**-,*On the decomposition of quadratic forms in six variables*, Acta Arith.**3**(1939), 64-78.**[12]**J.L. Lagrange, Oeuvres**3**(1869), 189-201.**[13]**O.T. O'Meara,*The integral representations of quadratic forms over local fields*, Amer. J. Math.**80**(1958), 843-878. MR**20:4526****[14]**-,*Introduction to quadratic forms*, Springer-Verlag, 1973.**[15]**L.J. Mordell,*A new Waring's problem with squares of linear forms*, Quart. J. Math. Oxford**1**(1930), 276-288.**[16]**S. Ramanujan,*On the expression of a number in the form*, Proc Cambridge Phil. Soc.**19**(1917), 11-21.**[17]**W. Plesken,*Additively indecomposable positive integral quadratic forms*, J. Number Theory**47**(1994), 273-283. MR**95c:11045****[18]**G.L. Watson,*The class number of a positive quadratic form*, Proc. London Math. Soc. (3)**13**(1963), 549-576. MR**27:107****[19]**-,*One-class genera of positive quadratic forms in at least five variables*, Acta Arith.**26**(1975), 309-327. MR**52:274****[20]**-,*One-class genera of positive quadratic forms in nine and ten variables*, Math.**25**(1978), 57-67. MR**58:10738****[21]**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**9:571e**

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Additional Information

**Byeong-Kweon Oh**

Email:
oandhan@math.snu.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-99-05254-5

Keywords:
$n$-universal lattice,
$U_{\mathbb{Z}}(n)$,
root lattice,
additively indecomposable

Received by editor(s):
April 27, 1998

Published electronically:
July 6, 1999

Additional Notes:
The author was partially supported by GARC and BSRI-98-1414

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 1999
American Mathematical Society