Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Orbits in unimodular Hermitian lattices


Author: Donald G. James
Journal: Trans. Amer. Math. Soc. 332 (1992), 849-860
MSC: Primary 11E39; Secondary 11E08, 11H06, 20G25, 20G30
MathSciNet review: 1089419
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ be a unimodular indefinite hermitian lattice over the integers $ \mathfrak{o}$ of an algebraic number field, and $ N(L,c)$ the number of primitive representations of $ c \in \mathfrak{o}$ by $ L$ that are inequalivant modulo the action of the integral special unitary group $ SU(L)$ on $ L$. The value of $ N(L,c)$ is determined from the local representations via a product formula.


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

  • [G] Larry J. Gerstein, Integral decomposition of hermitian forms, Amer. J. Math. 92 (1970), 398–418. MR 0269592
  • [Jc] Ronald Jacobowitz, Hermitian forms over local fields, Amer. J. Math. 84 (1962), 441–465. MR 0150128
  • [J1] D. G. James, On Witt's theorem for unimodular quadratic forms, Pacific J. Math. 26 (1968), 303-316 and 33 (1970), 645-652.
  • [J2] D. G. James, Representations by integral quadratic forms, J. Number Theory 4 (1972), 321–329. MR 0309871
  • [J3] D. G. James, Invariant submodules of unimodular Hermitian forms, Pacific J. Math. 72 (1977), no. 2, 471–482. MR 0562286
  • [J4] Donald G. James, Integral sums of squares in algebraic number fields, Amer. J. Math. 113 (1991), no. 1, 129–146. MR 1087804, 10.2307/2374824
  • [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. USSR-Izv. 14 (1980), 103-167.
  • [O'M] O. T. O'Meara, Introduction to quadratic forms, Springer-Verlag, New York, 1963.
  • [S] Goro Shimura, Arithmetic of unitary groups, Ann. of Math. (2) 79 (1964), 369–409. MR 0158882
  • [W] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328–338. MR 0138565

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11E39, 11E08, 11H06, 20G25, 20G30

Retrieve articles in all journals with MSC: 11E39, 11E08, 11H06, 20G25, 20G30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1089419-6
Keywords: Unimodular hermitian forms, algebraic integers, integral representations, unitary group, orbits, local fields, cyclotomic fields
Article copyright: © Copyright 1992 American Mathematical Society