Weil pairing and Galois descent

There is an interesting way in which Weil pairing on Abelian Varieties is nothing more than Galois descent for a particular Galois extension. I have not seen this connection used before in the literature but I have also not seen a lot of literature…

Let A/k be an abelian variety over a field k of characteristic p. Suppose m is an integer coprime to p and let [m]: A \to A denote the multiplication by m map with kernel A[m].

Recall that there is a dual abelian variety A^\vee representing the Picard functor for A. In particular, A^\vee[m] is the group of line bundles \mathcal L on A such that [m]^*\mathcal L is trivial. A and A^\vee are finite abelian groups and the Weil pairing is a perfect pairing of the form:

\langle-,-\rangle:  A[m] \times A^\vee[m] \to \mu_m.

It’s definition goes as follows: Let D be a divisor corresponding to \mathcal L and let g(x) be a rational function on A such that \mathrm{div}\ g(x) = [m]^*D. Then, for a \in A[m], we define \langle a,\mathcal L\rangle = g(x+a)/g(x).

If t_a denotes translation by a, then g(x+a) is the pull back of g(x) along t_a. Since [m]\circ t_a = [m], g(x+a) and g(x) have the same divisor and so g(x+a)/g(x) is regular everywhere on A and hence constant by properness of A. A little more work shows that this rational functions is in \mu_m. We will see below that this is an immediate conclusion of our alternate viewpoint.

This is the standard picture. However, there is also this alternate way of looking at things:

Consider the etale Galois extension [m]: A \to A. It’s galois group is canonically identified with A[m]. Moreover, A^\vee[m] is precisely the set of line bundles on A trivialized by this etale cover. In other words, it is the set of Galois twists of \mathcal O_A (or even for any line bundle). Since descent is effective for this extension, this will immediately imply that:

A^\vee[m] \cong H^1(\mathrm{Gal}([m]), \mathcal O_A^\times) = H^1(A([m]), k^\times) = \mathrm{Hom}\ (A[m],k^\times).

The final equality is because A[m] acts trivially on \mathcal O_A^\times = k^\times. We also see immediately that \mathrm{Hom}\ (A[m],k^\times) = \mathrm{Hom}\ (A[m],\mu_m) since A[m] is m-torison as an Abelian group. Therefore, this gives us a pairing:

A[m]\times A^\vee[m] \to \mu_m.

Explicitly, the map goes as follows: Given a line bundle \mathcal L, we pick an isomorphism g: O_A \to [m]^*\mathcal L. Then, the corresponding 1-cocycle for a \in A[m] is defined by a \to (t_a^*g)^{-1}g \in \mathrm{Aut}(\mathcal O_A). It is easily seen that this is the same explicit construction as in the standard viewpoint.

Conclusion:

I find the Galois descent viewpoint conceptually satisfying. The standard treatments of the Weil pairing can seem arbitrary and it not clear why such a pairing should exist or be useful. On the other hand, [m]:A \to A is a perfectly natural Galois extension connected to A and \mathcal O_A torsors are clearly an interesting thing to consider.

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