1 and λ on the two sheets of the Riemann surface. From the Seifert–van Kampen theorem, the homology group of the curve is free of rank two. Because the curves meet in a single point, , neither of their homology classes is a proper multiple of some other homology class, and hence they form a basis of H1. The period matrix for this family is therefore The first entry of this matrix we will abbreviate as A, and the second as B. The bilinear form Q is positive definite because locally, we can always write ω as f dz, hence By Poincaré duality, γ and δ correspond to cohomology classes γ* and δ* which together are a basis for . It follows that ω can be written as a linear combination of γ* and δ*. The coefficients are given by evaluating ω with respect to the dual basis elements γ and δ: When we rewrite the positive definiteness of Q in these terms, we have Since γ* and δ* are integral, they do not change under conjugation. Furthermore, since γ and δ intersect in a single point and a single point is a generator of H0, the cup product of γ* and δ* is the fundamental class of X0. Consequently this integral equals . The integral is strictly positive, so neither A nor B can be zero. After rescaling ω, we may assume that the period matrix equals for some complex number τ with strictly positive imaginary part. This removes the ambiguity coming from the action. The action of is then the usual action of the modular group on the upper half-plane. Consequently, the period domain is the Riemann sphere. This is the usual parameterization of an elliptic curve as a lattice. See also Hodge theory Jacobian variety Modular group References Calculations Explicit calculation of period matrices for curves of the form - includes examples Explicit calculation of period matrices for hyperelliptic curves - includes examples Algorithm for computing periods of hypersurfaces General Voisin, Hodge Theory and Complex Algebraic Geometry I, II Applications Shimura curves within the locus of hyperelliptic Jacobians in genus three External links Period mapping in the Encyclopedia of Mathematics Hodge theory Elliptic curves