Image courtesy of Andrew Snowden under a Creative Commons license: BY.
Math 679 is a graduate level mathematics course whose purpose is to prove Mazur's theorem. Mazur's theorem is a well-known and important result, however it is not often taught in classroom settings. The course is divided into three parts: elliptic curves and abelian varieties, moduli of elliptic curves, and proof of Mazur’s theorem.
Schedule
Lecture 1: Overview
Part I: Elliptic curves and abelian varieties
Lectures 2, 3, 4: Elliptic curves and abelian varieties over fields
Lectures 5, 6, 7: Group schemes, over fields and DVRs, including Raynaud’s theorem
Lectures 8, 9: Abelian varieties in mixed characteristic, including Néron models
Lecture 10: Jacobians
Lecture 11: Criterion for rank 0 (Theorem B from Lecture 1)
Part II: Moduli of elliptic curves
Lectures 12, 13, 14: Modular curves
Lecture 15, 16: Modular forms and the Hecke algebra
Lecture 17: The Eichler–Shimura theorem
Lectures 18, 19: Criterion for non-existence of torsion (Theorem A from Lecture 1)
Part III: Proof of Mazur’s theorem
Lecture 20: The Eisenstein ideal and Eisenstein quotient of J0(N)
Lectures 21, 22: The special fiber at N of J0(N)
Lecture 23: Ogg’s theorem on the order of [∞]−[0] in J0(N)
Lectures 24, 25, 26, 27: TBA
About the Creators
Andrew Snowden
Andrew Snowden is an assistant professor of mathematics at the University of Michigan.
Image courtesy of Andrew Snowden under a Creative Commons license: BY.
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Lecture 01: Exercise 1 |
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Lecture 01: Overview |
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Lecture 02: Elliptic curves |
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Lecture 03: Abelian varieties (analytic theory) |
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Lecture 04: Abelian varieties (algebraic theory) |
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Lecture 05: Group schemes 1 |
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Lecture 06: Group schemes 2 |
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Lecture 07: Group schemes 3 |
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Lecture 08: Elliptic curves over DVRs |
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Lecture 09: Néron models |
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Lecture 10: Jacobians |
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Lecture 11: Criterion for rank 0 |
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| Document Title | Creator | Downloads | License |
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Lecture 01: Overview |
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Lecture 02: Elliptic curves |
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Lecture 03: Abelian varieties (analytic theory) |
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Lecture 04: Abelian varieties (algebraic theory) |
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Lecture 05: Group schemes 1 |
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Lecture 06: Group schemes 2 |
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Lecture 07: Raynaud’s theorem |
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Lecture 08: Elliptic curves over DVRs |
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Lecture 09: Néron models |
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Lecture 10: Jacobians |
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Lecture 11: Criterion for rank 0 |
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Lecture 12: Modular curves over C |
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Lecture 13: Modular forms |
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Lecture 14: Lecture 14: Modular curves over Q |
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Lecture 15: Modular curves over Z |
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Lecture 16: Structure of the Hecke algebra |
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Lecture 17: Eichler–Shimura |
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Lecture 18: Criterion for non-existence of torsion points |
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Lecture 19: J0(N) mod N |
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Lecture 20: Proof of Mazur’s theorem (part 1) |
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Lecture 21: Proof of Mazur’s theorem (part 2) |
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Lecture 22: 13 torsion |
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Lecture 23: Finishing up |
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References |
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Plan |
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Image courtesy of Andrew Snowden under a Creative Commons license: BY.
Jump to:
| Document Title | Creator | Downloads | License |
|---|---|---|---|
|
Lecture 01: Overview |
Andrew Snowden
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Lecture 02: Elliptic curves |
Andrew Snowden
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Lecture 02: Elliptic curves |
Andrew Snowden
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Lecture 03: Abelian varieties (analytic theory) |
Andrew Snowden
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Lecture 03: Abelian varieties (analytic theory) |
Andrew Snowden
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Lecture 04: Abelian varieties (algebraic theory) |
Andrew Snowden
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Lecture 05: Group schemes 1 |
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Lecture 05: Group schemes 1 |
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Lecture 06: Group schemes 2 |
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Lecture 06: Group schemes 2 |
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Lecture 07: Raynaud’s theorem |
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Lecture 08: Elliptic curves over DVRs |
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Lecture 08: Elliptic curves over DVRs |
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Lecture 09: Néron models |
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Lecture 10: Jacobians |
Andrew Snowden
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Lecture 10: Jacobians |
Andrew Snowden
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Lecture 11: Criterion for rank 0 |
Andrew Snowden
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Lecture 11: Criterion for rank 0 |
Andrew Snowden
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Lecture 12: Modular curves over C |
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Lecture 13: Modular forms |
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Lecture 14: Lecture 14: Modular curves over Q |
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Lecture 15: Modular curves over Z |
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Lecture 16: Structure of the Hecke algebra |
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Lecture 17: Eichler–Shimura |
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Lecture 18: Criterion for non-existence of torsion points |
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Lecture 19: J0(N) mod N |
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Lecture 20: Proof of Mazur’s theorem (part 1) |
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Lecture 21: Proof of Mazur’s theorem (part 2) |
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Lecture 22: 13 torsion |
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Lecture 23: Finishing up |
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