Martin's Blog

Period relations on abelian varieties

Posted by Martin Orr on Monday, 26 October 2015 at 11:00

The Legendre period relation is a classical equation relating the periods and quasi-periods of an elliptic curve, as defined last time. I will discuss this relation, and then more generally discuss how the existence of polarisations implies relations between the periods of higher-dimensional abelian varieties.

These examples motivate the introduction of the geometric motivic Galois group, which gives an upper bound for the transcendence degree of periods of an abelian variety (or indeed any algebraic variety). This upper bound is conjectured to be equal to the actual transcendence degree. I had intended to discuss the geometric motivic Galois group in this post too, but I decided that it was getting to long so I will postpone that to another time.

The Legendre period relation

Let E be an elliptic curve over \mathbb{C} with equation  y^2 = x^3 + 4ax + b. Recall that there is a basis \{ \omega_1, \omega_2 \} for H^1_{dR}(E/\mathbb{C}) is represented by the differential forms of the second kind dx/y (which is regular) and x \, dx/y (which has a double pole at infinity). If we choose a basis \alpha, \beta for H_1(E(\mathbb{C}), \mathbb{Z}), then we can define the fundamental periods of E as  \lambda_1 = \int_\alpha \frac{dx}{y}, \; \lambda_2 = \int_\beta \frac{dx}{y} and the quasi-periods as  \eta_1 = \int_\alpha \frac{x \, dx}{y}, \; \eta_2 = \int_\beta \frac{x \, dx}{y}.

The Legendre period relation asserts that  \lambda_1 \eta_2 - \lambda_2 \eta_1 = 2\pi i. In the language of the previous post, the determinant of the extended period matrix of E is 2 \pi i. Note that the sign in this equation (2\pi i or -2\pi i) depends on the ordering of \alpha and \beta - this is chosen based on the standard orientation of E(\mathbb{C}) to ensure that we end up with 2 \pi i).

Following the introduction to Deligne's paper on absolute Hodge classes, I want to give a simple proof that if E is defined over the field k \subset \mathbb{C}, then the Legendre period relation holds up to multiplication by a scalar in k:  \lambda_1 \eta_2 - \lambda_2 \eta_1 \in 2\pi i \cdot k^\times. Note that if what we are really interested in is transcendence properties of the periods and quasi-periods, then an identity which holds up to multiplication by k is as good as an exact identity.

The key point is that the de Rham cohomology classes represented by dx/y and x \, dx/y are defined over k. Hence the extended period matrix  \Omega = \begin{pmatrix} \lambda_1 & \eta_1 \\ \lambda_2 & \eta_2 \end{pmatrix} expresses a basis for H^1_{dR}(E/k) in terms of a basis for H^1(E(\mathbb{C}), \mathbb{Q}), via the standard comparison isomorphism  H^1_{dR}(E/k) \otimes_k \mathbb{C} \cong H^1(E(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q} \mathbb{C}.

Since H^2(E) = \bigwedge^2 H^1(E), both in de Rham cohomology and in singular cohomology, it follows that \det \Omega is the coordinate for the basis element \omega_1 \smile \omega_2 \in H^2(E/k) relative to the basis element \alpha^\vee \smile \beta^\vee \in H^2(E(\mathbb{C}), \mathbb{Q}).

But there is a pair of bases for H^2(E/k) and H^2(E(\mathbb{C}), \mathbb{Q}) which we already know how to compare. We can take the cycle class of a point in each cohomology theory, and we have  cl_{dR}(pt) = 2 \pi i \cdot cl_{top}(pt).  \tag{*} Recall that we proved this relation for \mathbb{P}^1, and it motivates the definition of Tate twists of Hodge structures. We can deduce that the same relation holds on an elliptic curve (or indeed any smooth projective curve) by considering a finite morphism \pi \colon E \to \mathbb{P}^1, say of degree d. The pullback of a point in \mathbb{P}^1 is d points in E, and so  \pi^* cl_{dR,\mathbb{P}^1}(pt) = d \cdot cl_{dR,E}(pt) and similarly for cl_{top}. Dividing by d, the relation (*) for \mathbb{P}^1 implies the same relation for E.

Since \omega_1 \smile \omega_2 \in k^\times \cdot cl_{dR}(pt) and \alpha \smile \beta \in \mathbb{Q}^\times \cdot cl_{top}(pt), we conclude that  \lambda_1 \eta_2 - \lambda_2 \eta_1 \in 2\pi i \cdot k^\times.

Observe that \alpha^\vee \smile \beta^\vee is a generator for the \mathbb{Z}-module H^2(E(\mathbb{C}), \mathbb{Z}) and so in fact  \alpha^\vee \smile \beta^\vee = cl_{top}(pt). Thus showing that the Legendre period relation holds exactly is equivalent to showing that  \omega_1 \smile \omega_2 = cl_{dR}(pt). This is a purely algebraic result (it no longer involves integration as does the classical statement of the period relation). But I think that a purely algebraic proof of it is hard. Deligne sketches an analytic proof of this algebraic result.

Algebraic cycles and relations between periods

At first sight it might look like the Legendre period relation provides a lower bound for the transcendence degree of the periods of an elliptic curve - it implies that  \operatorname{trdeg} k(\lambda_1, \lambda_2, \mu_1, \mu_2)/k \geq 1 unless \pi \in \bar{k}. But really one should think of it as providing an upper bound for the transcendence degree over the field k(\pi):  \operatorname{trdeg} k(\pi, \lambda_1, \lambda_2, \mu_1, \mu_2)/k(\pi) \leq 3. Indeed, it does not just tell us that there exists an algebraic relation between the periods and quasi-periods over k(\pi), it tells us exactly what form that relation takes.

The presence of \pi in the field generated by periods is inevitable due to the issue of Tate twists. A similar argument to the argument for the Legendre period relation shows that, for any abelian variety of dimension g defined over k, the determinant of the extended period matrix is in (2 \pi i)^g \cdot k^\times.

The Legendre period relation and the more general relation for the determinant of the extended period matrix are examples of the principle that algebraic cycles on a variety imply algebraic relations between the periods. In particular, the determinant relation comes from the fact that  cl_{dR}(pt) = (2 \pi i)^g \cdot cl_{top}(pt) \in H^{2g}(A, \mathbb{C}).

For another example of relations between periods implied by an algebraic cycle, consider a polarisation on the abelian variety A. A polarisation is a symplectic pairing on H_1(A(\mathbb{C}), \mathbb{Z}) or equivalently an element of H^2(A(\mathbb{C}), \mathbb{Z}). The definition of polarisation requires that it must be of the form cl_{top}(D) for an ample divisor D on A.

The algebraic cycle D will also induce an element  cl_{dR}(D) \in H^2_{dR}(A/\bar{k}) and hence a symplectic pairing on H^1_{dR}(A/\bar{k}). The compatibility of the cycle class maps implies that under the comparison isomorphism H^2_{dR}(A/\mathbb{C}) \cong H^2(A(\mathbb{C}), \mathbb{C}), we get  cl_{dR}(D) = 2 \pi i \cdot cl_{top}(D). Now if we choose symplectic bases for H^1(A(\mathbb{C}), \mathbb{Q}) and H^1_{dR}(A/\bar{k}) with respect to the respective symplectic forms induced by D, and use these bases to calculate the extended period matrix, then we will get a matrix in the general symplectic group \operatorname{GSp}_{2g}(\mathbb{C}). This implies that the transcendence degree over k of the extended period matrix must be at most  \dim \operatorname{GSp}_{2g} = 2g^2 + g + 1, and for g > 1 this is better than the trivial upper bound of 4g^2.

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


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