Math 679 is a graduate level mathematics course whose purpose is to prove Mazur's theorem. Mazur's theorem is a wellknown 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 nonexistence 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.
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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 

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 nonexistence 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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Jump to:
Document Title  Creator  Downloads  License 

Lecture 01: Overview 
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Lecture 02: Elliptic curves 
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Lecture 02: Elliptic curves 
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Lecture 03: Abelian varieties (analytic theory) 
Andrew Snowden


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 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 
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Lecture 10: Jacobians 
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Lecture 11: Criterion for rank 0 
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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 nonexistence 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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