Each inclusion of open subgroups $H\leq G$ of $\GL_2(\widehat\Z)$ induces a morphism of modular curves $X_H\to X_G$: every elliptic curve with level $H$ structure gives rise to an elliptic curve with level $G$ structure. We call such a morphism of modular curves a **modular cover**.

The **degree** of the morphism $X_H\to X_G$ is equal to the ratio of the $\PSL_2$-index of $H$ by that of $G$. Note that when $-1$ is in $G$ but not $H$, we might have $H < G$ strictly contained but the map $X_H \to X_G$ may have degree $1$ (hence an isomorphism).

A modular cover is **minimal** if $H < G$ is a maximal (proper) subgroup.

The morphism $X_H\to X_G$ induces a surjective homomorphism $\phi \colon \operatorname{Jac}(X_H)\to\operatorname{Jac}(X_G)$. The **kernel** of the modular cover is the connected component of $\ker \phi$. This is an abelian variety, possibly of dimension zero.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by John Voight on 2022-11-05 16:55:12

**Referred to by:**

- modcurve.quadratic_refinements
- lmfdb/modular_curves/main.py (line 812)
- lmfdb/modular_curves/main.py (line 818)
- lmfdb/modular_curves/templates/modcurve.html (line 307)
- lmfdb/modular_curves/templates/modcurve.html (lines 358-369)
- lmfdb/modular_curves/templates/modcurve.html (lines 394-405)
- lmfdb/modular_curves/templates/modcurve.html (lines 430-441)

**History:**(expand/hide all)

- 2022-11-05 16:55:12 by John Voight
- 2022-03-25 00:57:34 by Bjorn Poonen
- 2022-03-25 00:55:04 by Bjorn Poonen
- 2022-03-24 18:27:45 by Bjorn Poonen
- 2022-03-20 21:16:00 by Andrew Sutherland
- 2022-03-20 17:29:00 by Andrew Sutherland

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