Lattices and number fields. (English) Zbl 0951.11016

Pragacz, Piotr (ed.) et al., Algebraic geometry: Hirzebruch 70. Proceedings of the algebraic geometry conference in honor of F. Hirzebruch’s 70th birthday, Stefan Banach International Mathematical Center, Warszawa, Poland, May 11-16, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 241, 69-84 (1999).
The paper deals with various aspects of integral symmetric bilinear forms \(b\) arising from a fractional ideal \(I\) of an algebraic number field \(K\) and some \(a \in K\) fixed by an involution \(z \mapsto \overline{z}\); in this situation \(b(x,y)\) for \(x,y \in I\) is defined as the trace of \(a x \overline{y}\). The author gives an overview of what is known about the forms arising from a given \(K\) (Section 2), and about fields leading to especially interesting positive definite lattices (Section 3). A connection between the property of \(K\) to be norm-euclidean and the covering radii of its ideal lattices is put into a general setting (Section 4). Motivated by a recent development in information theory, Section 5 discusses the so-called diversity for ideal lattices; this is defined with respect to a fixed orthonormal base of the ambient real vector space, and corresponds to the Hamming minimal weight for codes. Finally (Section 6), the role played by definite lattices in knot theory is touched upon.
For the entire collection see [Zbl 0924.00033].


11E12 Quadratic forms over global rings and fields
11H31 Lattice packing and covering (number-theoretic aspects)
57M25 Knots and links in the \(3\)-sphere (MSC2010)