GRAPHS, REPRESENTATIONS, AND SPINOR GENERA

A fundamental problem in the theory of quadratic forms is to find reasonable invariants that classify forms up to integral equivalence. A principal objective of this dissertation is to contribute to the solution of this problem by providing a method for determining whether two forms in three or more variables over the ring of integers of an algebraic number field are properly spinor equivalent.;Theorem: Given lattices L and M that are everywhere locally integrally equivalent, there exists a lattice M' (epsilon) spn('+)M and a prime such that M' (epsilon) R(L, ).;The proof of this theorem provides a method for finding such a prime . Together with an idelic criterion for determining whether R(L, ) contains lattices from one or two proper spinor genera, this allows one to determine whether M (epsilon) spn('+)L. The relationship of this criterion to a theorem of Cassels ({C}, {C(,1)}) is discussed. We also establish a relationship between this criterion and the theory of spinor exceptional integers.;In addition, for a ternary lattice L and a vector y (epsilon) L, we determine the shape of the subgraph R(,y)(L, ) whose vertices are lattices in R(L, ) which contain y. We use this to obtain some representation-theoretic results for ternary lattices over the ring of integers of an a algebraic number field analogous to results of Schulze-Pillot for ternary Z-lattices ({SP}, {SP(,1)}).;This method arises from a study of a graph R(L, ), where may be almost any prime, whose vertices are lattices M with M(, ) (TURNEQ) L(, ) and M(, ) = L(, ) for all primes (NOT=) . Generalizing a result of Schulze-Pillot ({SP}, {SP(,1)}) for ternary Z-lattices, we show that R(L, ) contains lattices from at most two proper spinor genera. The crux of our method is the following.;In a final chapter, we discuss ternary lattices that represent every integer permitted by spinor-genus considerations. Using a method of Watson ({W(,1)}), we obtain examples of such "spinor regular" lattices.;An Appendix contains a computer program for generating graphs for ternary Z-lattices. An Epilogue suggests several possible avenues for further research on the application of graphs to quadratic form theory.