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For the big modular curve $X(N)^{\text{big}}_{\Q}$, the set $\pi_0(X(N)^{\text{big}}_{\overline{\Q}})$ of geometric components is in bijection with the set $\mu_n^{\text{prim}}$ of primitive $N$th roots of unity (map $(E,P,Q)$ to the Weil pairing $e_N(P,Q)$).

For the quotient curve $(X_H)_\Q$, we have $\pi_0((X_H)_{\overline{\Q}}) \simeq \det(H_N) \backslash \pi_0(X(N)^{\text{big}}_{\overline{\Q}})\simeq \det(H_N) \backslash \mu_N^{\text{prim}}$. Thus the number of geometric components is the index $((\Z/N\Z)^\times : \det H_N)$.

The statements above hold also if $\Q$ and $\overline{\Q}$ are replaced by $k$ and $\overline{k}$, for any field $k$ of characteristic not dividing $N$.

The curve $(X_H)_{\Q}$ is integral. It is geometrically integral if and only if $\det H_N = (\Z/N\Z)^\times$, or equivalently $\det H = \widehat\Z^\times$.

Knowl status:
  • Review status: reviewed
  • Last edited by Bjorn Poonen on 2022-03-24 20:07:20
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