Let $N \ge 1$. Let $\mu_N$ be the group of $N$th roots of unity in some algebraically closed field of characteristic not dividing $N$. Let $M$ be a free rank $2$ $\Z/N\Z$-module together with an isomorphism $\alpha \colon \bigwedge^2 M \stackrel{\sim}\to \mu_N$, or equivalently with a nondegenerate alternating pairing $M \times M \to \mu_N$. For example, $M$ could be $E[N]$ for an elliptic curve $E$, together with the Weil pairing. Or $M$ could be $\Z/N\Z \times \mu_N$ with the "determinant" pairing $(a,\gamma),(b,\delta) \mapsto \delta^a/\gamma^b$.

A **symplectic isomorphism** from $M$ to another such structure $M'$ is a $\Z/N\Z$-module isomorphism $M \to M'$ such that the induced isomorphism $\bigwedge^2 M \to \bigwedge^2 M'$ gets identified via $\alpha$ and $\alpha'$ with the *identity* $\mu_N \to \mu_N$.

The same definition makes sense in a context in which each free rank 2 $\Z/N\Z$-module is enriched with a Galois action to make a Galois module, or replaced by a finite étale group scheme that is $(\Z/N\Z)^2$ étale locally.

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- Review status: beta
- Last edited by Bjorn Poonen on 2022-03-25 19:21:28

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