Martin's Blog

Periods of abelian varieties

Posted by Martin Orr on Tuesday, 06 October 2015 at 16:10

There are a couple of different matrices associated with an abelian variety which are referred to as its period matrix. These matrices relate different choices of bases for the tangent space or H^1 of the abelian variety. In this post I will discuss the different definitions of period matrices and how they relate to each other.

A complex abelian variety A can be realised as the quotient of the g-dimensional complex vector space T_0 A by the rank-2g lattice H_1(A, \mathbb{Z}). The period matrix expresses a basis of this lattice in terms of a basis for the tangent space. We can also get a period matrix which is twice as large by using de Rham cohomology H^1_{dR}(A) instead of the tangent space T_0 A.

The period matrix can be defined for any complex abelian variety, but it contains additional information if the abelian variety is defined over a number field k, as we can then choose our basis for T_0 A to also be defined over k. The period matrix then gives us a set of complex numbers relating a basis of a k-vector space to a basis of a \mathbb{Q}-vector space, and the transcendence properties of these numbers are interesting and I will discuss them in a later post.

The period lattice

Let's work first over the complex numbers. Let A be a complex abelian variety of dimension g. Classically, a period lattice of A is defined to be a lattice \Lambda \subset \mathbb{C}^g such that A \cong \mathbb{C}^g/\Lambda.

Recall that the lattice \Lambda is the kernel of the exponential map \exp_A \colon T_0 A \to A. This is where the name period lattice comes from: \Lambda consists of the periods if \exp_A in the sense that  \exp_A(z) = \exp_A(z + \lambda) \text{ for all } z \in \mathbb{C}^g \text{ iff } \lambda \in \Lambda. In order to identify \ker \exp_A with a lattice in \mathbb{C}^g, we have to choose a basis for T_0 A. Thus the period lattice is not uniquely defined, but only up to a change of basis for T_0 A, that is, up to multiplication by an element of \operatorname{GL}_g(\mathbb{C}).

For example, the period lattice of an elliptic curve is a rank-2 lattice in \mathbb{C}, defined up to multiplication by an element of \mathbb{C}^\times.

Fundamental periods

Rather than an infinite object like a lattice, it can be useful to represent the same information in a finite list of numbers. We can do this by choosing a basis for \Lambda. Then the coordinates of this basis (relative to our chosen basis for T_0 A) form a g \times 2g complex matrix. The basis vectors are called fundamental periods and the matrix of their coordinates is called a period matrix for A.

It is often helpful to restrict our choice of basis for \Lambda to only allow bases which are symplectic with respect to a chosen polarisation of A. Recall that a polarisation is a symplectic form  \psi \colon H_1(A, \mathbb{Z}) \times H_1(A, \mathbb{Z}) \to \mathbb{Z} satisfying some conditions. We can identify \Lambda = \ker \exp_A with H_1(A, \mathbb{Z}). A symplectic basis means a basis \alpha_1, \dotsc, \alpha_g, \beta_1, \dotsc, \beta_g for H_1(A, \mathbb{Z}) such that  \psi(\alpha_i, \beta_i) = 1 \text{ for all } i \text{ and } \psi(\alpha_i, \alpha_j) = \psi(\beta_i, \beta_j) = \psi(\alpha_i, \beta_j) = 0 \text{ for all } i \neq j.

Once we have fixed the polarisation, there are two ambiguities in the definition of the period matrix of a polarised abelian variety: we can multiply by an element of \operatorname{GL}_g(\mathbb{C}) on the left (changing the basis for T_0 A) and by an element of \operatorname{Sp}_{2g}(\mathbb{Z}) on the right (changing the symplectic basis for H_1(A, \mathbb{Z})).

The Siegel matrix

The Siegel matrix of a complex abelian variety A is obtained by choosing our basis for T_0 A to be the same as the first half \alpha_1, \dotsc, \alpha_g of our symplectic basis for H_1(A, \mathbb{Z}) (it is a theorem that these vectors are \mathbb{C}-linearly independent). Thus, the first g columns of the period matrix simply form the identity matrix, and the Siegel matrix is defined to be the second g columns of the period matrix.

Thus the Siegel matrix is a g \times g complex matrix. It is defined up to an action of \operatorname{Sp}_{2g}(\mathbb{Z}) (changing the symplectic basis for H_1(A, \mathbb{Z})) given by  \begin{pmatrix} A & B \\ C & D \end{pmatrix} Z = (AZ + B)(CZ + D)^{-1}. The Riemann bilinear relations assert that the Siegel matrix is symmetric and has positive definite imaginary part (i.e. it lives in the Siegel upper half-space \mathcal{H}_g). This leads to the construction of the moduli space of principally polarised abelian varieties as \operatorname{Sp}_{2g}(\mathbb{Z}) \backslash \mathcal{H}_g.

Note: the Siegel matrix is sometimes called the Riemann matrix and that is perhaps more sensible, but the term Riemann matrix often refers to the period matrix above, so I have chosen the name of Siegel matrix to avoid ambiguity.

Arithmetic information

Now let's suppose that our abelian variety is defined over a number field k. The tangent space T_0 A is a k-vector space, and so we can choose a k-basis for it and use that basis when computing the period matrix as above. The lattice \Lambda embeds into T_0 A_{\mathbb{C}}, not T_0 A_k, and so the period matrix still has complex entries (they are usually transcendental).

Now it is defined up to multiplication by \operatorname{GL}_g(k) on the left and \operatorname{Sp}_{2g}(\mathbb{Z}) on the right. Note that the field generated (over k) by the entries of the period matrix is not changed by these ambiguities, and in particular the transcendence degree of the periods is a well-defined invariant of A which I will say more about in the next post.

Calculating the periods

This abstract definition of periods as the kernel of \exp_A \colon T_0 A_{\mathbb{C}} \to A(\mathbb{C}) is all very well, but can we give a more concrete description? We can do this using the fact that the "evaluation at 0" map  \Omega^1(A) \to (T_0 A)^\vee is an isomorphism when A is an abelian variety (over any field of characteristic zero). Here \Omega^1(A) is the space of regular differential 1-forms on A.

The choice of a basis for T_0 A induces a dual basis of (T_0 A)^\vee, and hence a basis \omega_1, \dotsc, \omega_g for \Omega^1(A). To calculate the coordinates of a period \gamma \in \Lambda, we simply have to evaluate the elements of the dual basis for (T_0 A)^\vee at \gamma. Passing through the isomorphisms (T_0 A)^\vee \cong \Omega^1(A) and \Lambda \cong H_1(A(\mathbb{C}), \mathbb{Z}), this means computing the integrals  \int_\gamma \omega_1, \dotsc, \int_\gamma \omega_g.

For example, for an elliptic curve defined by an equation  y^2 = x^3 + ax + b, the space of differential forms \Omega^1(A) is generated by the regular differential form dx/y and we can calculate the fundamental periods as  \lambda_1 = \int_\alpha \frac{dx}{y}, \quad \lambda_2 = \int_\beta \frac{dx}{y}.

De Rham cohomology and the extended period matrix

There is another definition commonly used for the term period matrix. This starts from the comparison isomorphism between de Rham cohomology and singular cohomology:  H^1_{dR}(A/k) \otimes_k \mathbb{C}  \cong  H^1_{dR}(A/\mathbb{C})  \cong  H^1(A(\mathbb{C}), \mathbb{C})  \cong  H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C}. We can choose a k-basis for H^1_{dR}(A/k) and a \mathbb{Q}-basis for H^1(A(\mathbb{C}), \mathbb{Q}), then write down the matrix for the above isomorphism in terms of these bases.

This gives a 2g \times 2g complex matrix, sometimes also called the period matrix. In order to avoid confusion I shall call this one the extended period matrix. It is defined up to multiplication by \operatorname{GL}_{2g}(\mathbb{Q}) on the left and by \operatorname{GL}_{2g}(k) on the right. Once again the field generated over k by the entries of the extended period matrix is independent of the choices of basis.

Relationship between the two period matrices

How do the period matrix and extended period matrix relate to each other? There is an injection  \Omega^1(A/k) \to H^1_{dR}(A/k) whose image is the component F^1 H^1_{dR}(A/k) of the Hodge filtration. This follows from the fact that that the Hodge filtration on the de Rham H^1 of an abelian variety matches up with the inclusion  T_0(A)^\vee \subset T_0(E_A)^\vee.

So we can take a basis for \Omega^1(A/k) and extend it to a basis for H^1_{dR}(A/k). We can also take a basis for H_1(A, \mathbb{Z}) and dualise to get a basis for H^1(A, \mathbb{Q}), and use these bases to write down the extended period matrix. Thus we see that, if we choose the bases appropriately, the g \times 2g period matrix defined using the period lattice is equal to the transpose of the left half of the 2g \times 2g extended period matrix. (The "transpose" comes from the fact that one was defined using homology, the other using cohomology.) Furthermore, the Hodge filtration of de Rham cohomology tells us which half of the extended period matrix to use.

Differential forms of the second kind

The entries of the extended period matrix which do not come from \Omega^1(A) are sometimes called quasi-periods. We would like to be able to compute these as integrals too. In order to do this, we introduce differential forms of the second kind (the regular differential forms considered before are sometimes called differential forms of the first kind).

Differential forms of the second kind are a kind of meromorphic differential forms. A meromorphic differential form is a differential forms which, locally at every point, looks like  \sum_i f_i dz_i where z_i are local coordinates and f_i are meromorphic functions of (z_1, \dotsc, z_n). (Meromorphic may be understood in terms of complex analysis or algebraic geometry as the context demands.) In other words, meromorphic differential forms are differential forms which are allowed to have poles.

If f is a meromorphic function, then df is a meromorphic differential forms. We call such forms exact meromorphic forms.

A differential form of the second kind is defined to be a meromorphic differential form which, locally at every point, is exact. In other words, the poles of a differential form of the second kind must "look like" poles of exact meromorphic forms.

For a more concrete description of differential forms of the second kind, let's restrict attention to curves. If \omega is a differential form on a curve C, we define the residue of \omega at a point P \in C by choosing a local coordinate z at P, writing out the Laurent expansion  \omega = \sum_{k = -n}^\infty a_k z^k dz and setting  \operatorname{Res}_P \omega = a_{-1}.

Observe that an exact meromorphic form has residue zero at every point (because, in order to get z^{-1} dz to appear in df, you would have to differentiate \log z but that is not meromorphic) and hence a differential form of the second kind has residue zero at every point. It turns out that this is the only restriction: a meromorphic differential form on a curve is of the second kind if and only if it has residue zero at every point.

An example of a differential form of the second kind on an elliptic curve which is not regular or exact is x \, dx/y. This has a pole at infinity and nowhere else, and in terms of a suitable local coordinate z at infinity it can be written as z^{-2} dz + something regular, so it has residue zero.

Computing the quasi-periods

The importance of differential forms of the second kind lies in the fact that H^1_{dR}(A/k) is isomorphic to the quotient  \{ \text{closed differential forms of the second kind} \} / \{ \text{exact meromorphic forms} \}.

Hence we can compute quasi-periods by integrating closed differential forms of the second kind in the same way as we compute periods by integrating regular differential forms.

For example, if A is the elliptic curve with equation y^2 = x^3 + ax + b, then the forms dx/y and x\,dx/y form a basis for H^1_{dR}(A/k) and we can calculate the quasi-periods as  \eta_1 = \int_\alpha \frac{x \, dx}{y}, \quad \eta_1 = \int_\beta \frac{x \, dx}{y}.

The Legendre period relation for elliptic curves

I have already mentioned that an elliptic curve has periods \lambda_1 and \lambda_2 obtained by integrating dx/y along generators for H_1(A, \mathbb{Z}) and quasi-periods \eta_1 and \eta_2 obtained by integrating x\,dx/y along the same paths. The Legendre period relation tells us that the extended period matrix has determinant 2\pi i, that is,  \lambda_1 \eta_2 - \lambda_2 \eta_1 = 2 \pi i. This post has got long enough without discussing that, so I will talk about it in the next post.

Tags abelian-varieties, alg-geom, hodge, maths, number-theory

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