On Minkowski reduction of positive quaternary quadratic forms

Period Matrices of Real Riemann Surfaces and Fundamental Domains

“…This explicit result is due to E.S. Barnes and M.J. Cohn (see [1]). We used it to determine a fundamental domain D for the congruence action of K 4,1 on Sym >0 (4, R).…”

“…, f mn are linear homogeneous expressions of the entries of σ. For low dimensions these expressions were explicitly determined: see [1] and [17]. After considerations of the previous sections, we now need a fundamental domain for the congruence action, on Sym >0 (2g 0 , R), of the group K 2g 0 ,1 , which is strictly contained in GL(2g 0 , Z) for 2g 0 ≥ 4.…”

“…, f mn are linear homogeneous expressions of the entries of σ. For low dimensions these expressions were explicitly determined: see [1] and [17].…”

“…where r denotes the morphism of the homology into itself induced by r. Given any Riemann surface (not necessarily a real one) with a fixed basis in the homology B, one can compute the corresponding period matrix 2 . This is a point in the Siegel upper half plane 1 In particular, this implies that the genus and the number of ovals of a separated, real Riemann surface need to satisfy the following relation: n ≡ g + 1 mod 2. For more details see [4,Theorem 1.5.3].…”

Minkowski's fundamental inequality for reduced positive quadratic forms

“…For n = 4, it is shown in Barnes and Cohn (1976) that ^ has 39 facets, which correspond to the 3 inequalities (4.1) a^a^a^^aâ nd all 36 inequalities of the form (2.1) for which x t = 1, x } = 0 if j> i, and the other Xj = 0 or +1 (excluding the 4 unit vectors). It appears to be computationally more economical to use Jt+ and then reject those notices of 2 + (a) which are not vertices of Si(a).…”