# Lattices Over Orders II by Klaus W. Roggenkamp

By Klaus W. Roggenkamp

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14), and M and M ~ are the same 01 = O2-module. Thus 32 vx 32 re= ~h~dQl(M) = E n d ~ ( M ~ ) =C~ and ~ = ~ , a contradiction. 8), the lattices {M~ IQ~ I~ . K Z S~ , ~ a T} are non-lsomorphlc irreducible A-lattlces! but this is the statement of ( i ) . (ii) If =I~(A ) = ~Jc T =-I~(V ~ ), then everYrlnglrreduclble A-lattlce M is of the form M~ ,Q~ I~ and thus EndA(M) is a maximal order in K ~ (cf. 5). ~ Endp (M~ ,Q~ I~ ) which Conversely, assume that ~M = EndA(M) is a maximal R-order for every M E¢ Ir(A ).

3) to conclude that A is reduclble! , f(X) is reducible. Thus f(X) must be irreducible modulo ~ S f o r sufficiently large s. ns K is an h-fleld with Dedeklnd domain R, A is a finite dimensional separable K-algebra and A an R-order in A. M • ^~o satisfies Elchler's condition! , none of the simple components in EndA(KM) is a totally definite quaternlon algebra. ma! ~-order. Let us assume that Elchler's theorem is true for maxlmal Rorders in A, and let r be a maximal R-order In A containing A. Then V M e r =M O also satisfies Elchler's condition, since E n d A ( K ~ M ) = = EndA(KM).

Then there exist integers ~Xl~l~l~, not all zero, such that n I i:i ajlxil • kj, anlX 1 ] ! kn. ,x n ) ~ ~R( n ) , the n-dimensional vectorspace over R, is called a lattice point if x i ~ =Z,l-~i~-n, and x i # 0 for at least one i. ,x n) and n LI(X) : ~Jffil aijxj" To prove Mlnkovski's lemma, we assume it to be false! , every lattice point x satisfies at least one of the Inequalities I LI(x) I ~ k I , l~l--n-l, Now, given a set {~i~lmi~_n of positive real numbers, show that there are only finitely many lattice points ~ satisfying ~Li(x)l< c~i.