Abstract
(Englisch)
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Determinants of integral ideal lattices and automorphisms of given characteristic polynomial
Let K be an algebraic number field endoved with a non-trivial involution - : K ® K. Let F be the fixed field of the involution. We denote by OK the ring of integers of K, and by OF the ring of integers of F . An integral ideal lattice is an integral lattice (I,b) , where I is a fractional OK-ideal, satisfying the relation
b(l x, y) = b( x, ly)
for all x,y I , and for all l OK.
It is easy to see that this is equivalent with the existence of an a K* with a = a such that
b(x,y ) = Tr (a x y),
for x,y I , where Tr = TrK/Q: K ® Q is the trace map.
It is natural to try to characterise the ideal lattices defined on a given algebraic number field; in particular, to ask which determinants and signatures are possible. We give such a characterisation in terms of ramification properties of the extension K/F . The details are given in [1].
A lattice is said to be unimodular if its determinant is ± 1 . In particular, we obtain a characterization of the signatures of unimodular lattices :
Proposition: Suppose that no prime OF-ideal ramifies in K/F. Let p, q be two non-negative integers with p + q = 2n . Then there exists a unimodular ideal lattice of signature (p,q) if and only if the following conditions hold :
(a) (p,q) ³ (n-s,n-s) and (p,q) º (n-s,n-s) (mod 2)
(b) p º q ( mod 8).
Integral ideal lattices occur in several applications, for instance to symmetric bilinear forms with an additional structure, like an automorphism with given characteristic polynomial. A recent paper of B. Gross and C. McMullen [2] deals with the characteristic polynomials of automorphisms of even, unimodular lattices with signature (p,q). Using the char-acterisation of signatures and determinants of ideal lattices, we give a different proof of their results, cf. [1].
[1] E. Bayer--Fluckiger, Determinants of integral ideal lattices and automorphisms of given characteristic polynomial, to appear.
[2] B.H. Gross and C.T. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, to appear.
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