Orbits in unimodular Hermitian lattices
Author:
Donald G. James
Journal:
Trans. Amer. Math. Soc. 332 (1992), 849860
MSC:
Primary 11E39; Secondary 11E08, 11H06, 20G25, 20G30
MathSciNet review:
1089419
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Abstract: Let be a unimodular indefinite hermitian lattice over the integers of an algebraic number field, and the number of primitive representations of by that are inequalivant modulo the action of the integral special unitary group on . The value of is determined from the local representations via a product formula.
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 L. J. Gerstein, Integral decomposition of hermitian forms, Amer. J. Math. 92 (1970), 398418. MR 0269592 (42:4487)
 [Jc]
 R. Jacobowitz, Hermitian forms over local fields, Amer. J. Math. 84 (1962), 441465. MR 0150128 (27:131)
 [J1]
 D. G. James, On Witt's theorem for unimodular quadratic forms, Pacific J. Math. 26 (1968), 303316 and 33 (1970), 645652.
 [J2]
 , Representations by integral quadratic forms, J. Number Theory 4 (1972), 321329. MR 0309871 (46:8976)
 [J3]
 , Invariant submodules of unimodular hermitian forms, Pacific J. Math. 72 (1977), 471482. MR 0562286 (58:27766)
 [J4]
 , Integral sums of squares in algebraic number fields, Amer. J. Math. 113 (1991), 129146. MR 1087804 (92j:11036)
 [K]
 D. Z. Kilhefner, Integral extensions of isometries of unimodular hermitian forms, Ph. D. Thesis, Pennsylvania State Univ., 1971.
 [N]
 V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSRIzv. 14 (1980), 103167.
 [O'M]
 O. T. O'Meara, Introduction to quadratic forms, SpringerVerlag, New York, 1963.
 [S]
 G. Shimura, Arithmetic of unitary groups, Ann. of Math. 79 (1964), 369409. MR 0158882 (28:2104)
 [W]
 C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328338. MR 0138565 (25:2009)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199210894196
PII:
S 00029947(1992)10894196
Keywords:
Unimodular hermitian forms,
algebraic integers,
integral representations,
unitary group,
orbits,
local fields,
cyclotomic fields
Article copyright:
© Copyright 1992 American Mathematical Society
