# Math 679 - Elliptic Curves

Image courtesy of Andrew Snowden under a Creative Commons license: BY.

Term:
Fall 2013
Published:
January 3, 2014
Revised:
June 5, 2015

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

## 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.

Term:
Fall 2013
Published:
January 3, 2014
Revised:
June 5, 2015

### Exercises

Lecture 01: Exercise 1

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### Handouts

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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### Lectures

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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### Schedules

Plan

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Image courtesy of Andrew Snowden under a Creative Commons license: BY.

Term:
Fall 2013
Published:
January 3, 2014
Revised:
June 5, 2015

### Part I: Elliptic curves and abelian varieties

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)

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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 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

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### Part II: Moduli of elliptic curves

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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### Part III: Proof of Mazur’s theorem

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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