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Modular curves of prime-power level with infinitely many rational points Sutherland, Andrew
Description
For each open subgroup $G$ of $\mathrm{GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants. This is joint work with David Zywina.
Item Metadata
Title |
Modular curves of prime-power level with infinitely many rational points
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-06-01T11:01
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Description |
For each open subgroup $G$ of $\mathrm{GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit
classification of the possible images of $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants. This is joint work with David Zywina.
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Extent |
50 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Massachusetts Institute of Technology
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Series | |
Date Available |
2017-11-29
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0360760
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Other
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International