Elliptic normal curves of even degree and theta functions (Q6573193)

Let \(N\) be an integer \(\geq 4\), and \(K\) a field of characteristic not dividing \(N\) with separable closure \(K_{\mathrm s}\). Then an elliptic curve \(E\) over \(K\) can be immersed in \({\mathbb P}^{N-1}\) as a curve of degree \(N\) by means of the linear system of \(|N \{O\}|\), where \(O\) is the origin of \(E\). A classical result going back to Bianchi and Klein states that if \(N\) is odd, this immersion is uniquely determined by specifying a full-level \(N\) structure, namely \(\Gamma(N)\)-structure which is fixing a basis \((S, T)\) of the group \(E[N]\) of \(N\)-torsion points in \(K_{\mathrm s}\) such that the Weil pairings \(e_{N}(S, T)\) become a fixed primitive \(N\)th root of \(1\).\N\NIn the present paper, the authors extend this result to the case when \(N\) is even by replacing \N\[\N\Gamma(N) = \left\{ \left. \left( \begin{array}{cc} a & b \\\Nc & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \left( \begin{array}{cc} a & b \\\Nc & d \end{array} \right) \equiv \left( \begin{array}{cc} 1 & 0 \\\N0 & 1 \end{array} \right) \mbox{mod} \ N \right\} \N\]\Nwith its subgroup \N\[\N\Gamma^{(N)}(2N) = \left\{ \left. \left( \begin{array}{cc} a & b \\\Nc & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \begin{array}{l} a \equiv d \equiv 1 \ \mbox{mod} \ N, \\\Nb \equiv c \equiv 0 \ \mbox{mod} \ 2N \end{array} \right\}.\N\]\NDenote by \(\mu_{N}\) the group of \(N\)th roots of \(1\) in \(K_{\mathrm s}\). Since a \(\Gamma(N)\)-structure is a symplectic isomorphism \({\mathbb Z}/N {\mathbb Z} \times \mu_{N} \rightarrow E[N]\), a \(\Gamma^{(N)}(2N)\)-structure above a \(\Gamma(N)\)-structure is defined as a pair of symplectic isomorphisms \(\phi_{N} : {\mathbb Z}/N {\mathbb Z} \times \mu_{N} \rightarrow E[N]\) and \(\phi_{2N} : {\mathbb Z}/2N {\mathbb Z} \times \mu_{2N} \rightarrow E[2N]\) satisfying \N\[\N[2] \left( \phi_{2N} (a \, \mbox{mod} \ 2N, \zeta) \right) = \phi_{N} (a \ \mbox{mod} \ N, \zeta^{2}) \N\]\Nfor any \((a \, \mbox{mod} \ 2N, \zeta) \in {\mathbb Z}/2N {\mathbb Z} \times \mu_{2N}\). Then the authors show that when \(N\) is even the immersion \(E \hookrightarrow {\mathbb P}^{N-1}\) is uniquely determined by specifying a \(\Gamma^{(N)}(2N)\)-structure above a \(\Gamma(N)\)-structure from which one obtain the universal elliptic curve over the modular curve associated with \(\Gamma^{(N)}(2N)\). Furthermore, they describe this immersion over \({\mathbb C}\) by \N\[\N\theta_{k}^{(N)}(z, \tau) = \theta_{(\frac{1}{2} - \frac{k}{N}, \frac{N}{2})} (N z, N \tau), \N\]\Nwhere \(\theta_{(p, q)}(z, \tau)\) denote the theta functions with characteristic \((p, q)\), and also quadratic equations satisfied by these theta functions.