Elliptic normal curves of even degree and theta functions. (English) Zbl 07881681
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 \(1\).
In the present paper, the authors extend this result to the case when \(N\) is even by replacing \[ \Gamma(N) = \left\{ \left. \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \equiv \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \mbox{mod} \ N \right\} \] with its subgroup \[ \Gamma^{(N)}(2N) = \left\{ \left. \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \begin{array}{l} a \equiv d \equiv 1 \ \mbox{mod} \ N, \\ b \equiv c \equiv 0 \ \mbox{mod} \ 2N \end{array} \right\}. \] Denote 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 \[ [2] \left( \phi_{2N} (a \, \mbox{mod} \ 2N, \zeta) \right) = \phi_{N} (a \ \mbox{mod} \ N, \zeta^{2}) \] for 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 \[ \theta_{k}^{(N)}(z, \tau) = \theta_{(\frac{1}{2} - \frac{k}{N}, \frac{N}{2})} (N z, N \tau), \] where \(\theta_{(p, q)}(z, \tau)\) denote the theta functions with characteristic \((p, q)\), and also quadratic equations satisfied by these theta functions.
In the present paper, the authors extend this result to the case when \(N\) is even by replacing \[ \Gamma(N) = \left\{ \left. \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \equiv \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \mbox{mod} \ N \right\} \] with its subgroup \[ \Gamma^{(N)}(2N) = \left\{ \left. \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL_{2}({\mathbb Z}) \ \right| \begin{array}{l} a \equiv d \equiv 1 \ \mbox{mod} \ N, \\ b \equiv c \equiv 0 \ \mbox{mod} \ 2N \end{array} \right\}. \] Denote 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 \[ [2] \left( \phi_{2N} (a \, \mbox{mod} \ 2N, \zeta) \right) = \phi_{N} (a \ \mbox{mod} \ N, \zeta^{2}) \] for 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 \[ \theta_{k}^{(N)}(z, \tau) = \theta_{(\frac{1}{2} - \frac{k}{N}, \frac{N}{2})} (N z, N \tau), \] where \(\theta_{(p, q)}(z, \tau)\) denote the theta functions with characteristic \((p, q)\), and also quadratic equations satisfied by these theta functions.
Reviewer: Takashi Ichikawa (Saga)
References:
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