Period Mapping - The Case of Elliptic Curves

The Case of Elliptic Curves

Consider the family of elliptic curves

where λ is any complex number not equal to zero or one. The Hodge filtration on the first cohomology group of a curve has two steps, F0 and F1. However, F0 is the entire cohomology group, so the only interesting term of the filtration is F1, which is H1,0, the space of holomorphic harmonic 1-forms.

H1,0 is one-dimensional because the curve is elliptic, and for all λ, it is spanned by the differential form ω = dx/y. To find explicit representatives of the homology group of the curve, note that the curve can be represented as the graph of the multivalued function

on the Riemann sphere. The branch points of this function are at zero, one, λ, and infinity. Make two branch cuts, one running from zero to one and the other running from λ to infinity. These exhaust the branch points of the function, so they cut the multi-valued function into two single-valued sheets. Fix a small ε > 0. On one of these sheets, trace the curve γ(t) = 1/2 + (1/2 + ε)exp(2πit). For ε sufficiently small, this curve surrounds the branch cut and does not meet the branch cut . Now trace another curve δ(t) that begins in one sheet as δ(t) = 1 + 2(λ − 1)t for 0 ≤ t ≤ 1/2 and continues in the other sheet as δ(t) = λ + 2(1 − λ)(t − 1/2) for 1/2 ≤ t ≤ 1. Each half of this curve connects the points 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, 1 + ε, 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 √(−1)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 H1(X₀, Z). 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 (1 τ) for some complex number τ with strictly positive imaginary part. This removes the ambiguity coming from the GL(1, C) action. The action of SL(2, Z) 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.