An exact mass formula for quadratic forms over number fields

Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

“…Algorithm 5.8 also works for indefinite lattices L, though in this case the mass of a genus Gen(L) is given as a sum over all class representatives L i of the covolumes Vol(Z/Aut O F (L i )) of the integral automorphism group Aut O F (L i ) with respect to some fixed measure on the symmetric space Z of the orthogonal group of Q (e.g. [18]). While these terms are computable in principle (by giving a presentation for Aut O F (L i ) and computing an explicit integral), this is not nearly as pleasant as counting the finitely many automorphisms arising in the totally definite case.…”

“…[4,5,6]), proving explicit enumerative results even in this simplified context has been a rather daunting endeavor due to their complexity and many opportunities for errors. A pioneer in these investigations has been Shimura, whose many papers [39,35,36,38] and recent book [37] focusing on the arithmetic of maximal lattices have set the stage for other authors' work [18,12,47,20,26]. Several other papers in a different style where maximal lattices play an important role are [1,29,30,8,46], and they are also mentioned in the introductory books [27, §82H and §104:9-10] and [13, §9.3].…”

The Structure of Masses of rank $n$ Quadratic Lattices of varying determinant over number fields

“…We can similarly use [14,Lemma 2.2] to express the lattice local densities β p (L, Q) in terms of the local densities the quadratic form ψ p induced by restricting Q to L p in some local basis of L p , giving…”