Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields
Jonathan Hanke · Contemporary mathematics - American Mathematical Society · 2013
In this paper we describe an algorithm that quickly computes a maximal a \mathfrak {a} -valued lattice in an F F -vector space equipped with a non-degenerate bilinear form, where a \mathfrak {a} is a fractional ideal in a number field F F . We then apply this construction to give an algorithm to compute an a \mathfrak {a} -maximal lattice in a quadratic space over any number field F F where the prime p = 2 p=2 is unramified. We also develop the theory of p \mathfrak {p} -neighbors for a \mathfrak {a} -valued quadratic lattices at an arbitrary prime p \mathfrak {p} of O F \mathcal {O}_F (including when p ∣ 2 \mathfrak {p}\mid 2 ) and prove its close connection to the residual geometry of certain quadrics mod p \mathfrak {p} . Finally we give a well-known application of p \mathfrak {p} -neighboring lattices and exact mass formulas to compute a complete set of representatives for the classes in a given genus of (totally definite) quadratic O F \mathcal {O}_F