The spinor genus of quaternion orders

Author:
Gordon L. Nipp

Journal:
Trans. Amer. Math. Soc. **211** (1975), 299-309

MSC:
Primary 10C05; Secondary 16A18

DOI:
https://doi.org/10.1090/S0002-9947-1975-0376526-6

MathSciNet review:
0376526

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Abstract: Let *D* be a global domain whose quotient field *F* does not have characteristic 2, let be a quaternion algebra over *F*, and let be an order on over *D*. A right -module *M* which is simultaneously a lattice on over *D* is said to be right -generic if there exists , such that . Our main result is that every right -generic module is cyclic if and only if every class in the spinor genus of represents a unit in *D*. One consequence is that is in a spinor genus of one class if and only if -generic modules are cyclic and represents every unit represented by its spinor genus. In addition, it is shown that a necessary and sufficient condition that an integral ternary lattice *L* be in a spinor genus of one class is that every right -generic pair be equivalent to a two-sided -generic pair, where is the quaternion order associated with *L*.

**[1]**M. Auslander and O. Goldman,*Maximal orders*, Trans. Amer. Math. Soc.**97**(1960), 1-24. MR**22**# 8034. MR**0117252 (22:8034)****[2]**L. E. Dickson,*Studies in the theory of numbers*, Chelsea, New York, 1930.**[3]**I. Kaplansky,*Submodules of quaternion algebras*, Proc. London Math. Soc. (3)**19**(1969), 219-232. MR**39**# 1496. MR**0240142 (39:1496)****[4]**G. Nipp,*Quaternion orders associated with ternary lattices*, Pacific J. Math.**53**(1974). MR**0357324 (50:9792)****[5]**O. T. O'Meara,*Introduction to quadratic forms*, Die Grundlehren der math. Wissenschaften, Band 117, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR**27**# 2485.**[6]**G. Pall,*On generalized quaternions*, Trans. Amer. Math. Soc.**59**(1946), 280-332. MR**8**, 318. MR**0018702 (8:318b)****[7]**G. L. Watson,*Integral quadratic forms*, Cambridge Univ. Press, Cambridge, 1960. MR**0118704 (22:9475)**

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0376526-6

Article copyright:
© Copyright 1975
American Mathematical Society