Math 525: Elliptic Curves

Problem Sets:

Texts:

[C]: J. W. S. Cassels, Lectures on Elliptic Curves. London Mathematical Society Student Texts 24. Cambridge University Press, 1991.
[S1]: Joseph Silverman, The Arithmetic of Elliptic Curves. Second Edition. Springer, 2009.
[S2]: Joseph Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994.

Schedule:

Date Topics References Notes

Mon 1/13

introduction, various definitions for elliptic curves 1/13 to 1/15

Wed 1/15 the Mordell–Weil group [C] §7,

Fri 1/17

algebraic sets, algebraic varieties, and dimension [S1] ch. I-II

1/17 to 1/24

Wed 1/22

singularities, rational maps and morphisms [S1] ch. I

Fri 1/24

valuations, maps on function fields, ramification [S1] ch. II

Mon 1/27 divisors and differentials [S1] ch. II 1/27 to 1/31

Wed 1/29 more divisors and differentials [S1] ch. II

Fri 1/31

Riemann–Roch, Weierstrass equations and genus \(1\) [S1] ch. II, ch. III §3

Mon 2/3

Riemann–Hurwitz, various Weierstrass equations [S1] ch. II, ch. III §1 2/3 to 2/7

Wed 2/5

j-invariants, singular Weierstrass cubics [S1] ch. III §1-2

Fri 2/7 the algebraic group law [S1] ch. III §3

Mon 2/10 isogenies [S1] ch. III §4 2/10 to 2/17

Wed 2/12 invariant differential [S1] ch. III §5

Fri 2/14 dual isogeny [S1] ch. III §6

Mon 2/17

dual isogeny, applications to torsion [S1] ch. III §6

Wed 2/19

Tate module, Galois representations [S1] ch. III §7

2/19

Fri 2/21 The Weil pairing [S1] ch. III §8 2/21 to 2/26

Mon 2/24

Weil pairing on Tate module, applications [S1] ch. III §8

Wed 2/26

Endomorphisms

[S1] ch. III §9-10

Fri 2/28

Endomorphisms in positive characteristic [S1] ch. V §3

2/28 and 3/5

Mon 3/3

Elliptic curves over \(\mathbb{C}\)

[S1] ch. VI, [S2] ch. I recording, notes

Wed 3/5 Hasse bound, Weil conjectures [S1] ch. V §1-2

Fri 3/7

Over local fields: reduction and torsion [S1] ch. VII §1-3

3/7 to 3/10

Mon 3/10 Action of inertia, reduction types [S1] ch. VII §4-5

Wed 3/12 Weak Mordell–Weil pt 1 [S1] ch. VIII §1 3/12 to 3/17

Fri 3/14 no class

Mon 3/17

Weak Mordell–Weil pt 2, Descent Theorem [S1] ch. VIII §1, 3

Wed 3/19

Mordell’s Theorem, heights on \(\mathbb{P}^N\) [S1] ch. VIII §4, 5 3/19 to 3/21

Fri 3/21 More heights [S1] ch. VIII §5

Spring Break

Mon 3/31 Heights on elliptic curves [S1] ch. VIII §6 3/31 to 4/2

Wed 4/2 Finishing MW! [S1] ch. VIII §6

Fri 4/4 Canonical Height [S1] ch. VIII §8 4/4

Mon 4/7

Diophantine Approximation

[S1] ch. VIII §1,2 4/7 to 4/9, Silverman’s comments on \(\nu\)-adic distance on curve

Wed 4/9 \(\nu\)-adic distance on a curve [S1] ch. VIII §2

Fri 4/11

applications of Siegel’s Theorem

[S1] ch. VIII §2-3 4/11 to 4/14

Mon 4/14 proof of Siegel’s Theorem [S1] ch. VIII §3

Wed 4/16 \(S\)-unit equations [S1] ch. VIII §4 4/16

Fri 4/18 Kummer via Galois cohomology [S1] ch. X §1 4/18-4/21

Mon 4/21

Computing the MW Group, \(2\)-descent [S1] ch. X §1

4/21

Wed 4/23 Twists of curves [S1] ch. X §2 4/25 and 4/30

Fri 4/25 Presentation – Chirag

Mon 4/30 Principal homogenous spaces [S1] ch. X §3

Wed 4/28 Presentation – Vibhu

Fri 5/2

Selmer and Shafarevich–Tate groups [S1] ch. X §4

5/2

Thurs 5/8 Presentations – Tanis, André, Julian

Date	Topics	References	Notes
Mon 1/13	introduction, various definitions for elliptic curves		1/13 to 1/15
Wed 1/15	the Mordell–Weil group	[C] §7,
Fri 1/17	algebraic sets, algebraic varieties, and dimension	[S1] ch. I-II	1/17 to 1/24
Wed 1/22	singularities, rational maps and morphisms	[S1] ch. I
Fri 1/24	valuations, maps on function fields, ramification	[S1] ch. II
Mon 1/27	divisors and differentials	[S1] ch. II	1/27 to 1/31
Wed 1/29	more divisors and differentials	[S1] ch. II
Fri 1/31	Riemann–Roch, Weierstrass equations and genus \(1\)	[S1] ch. II, ch. III §3
Mon 2/3	Riemann–Hurwitz, various Weierstrass equations	[S1] ch. II, ch. III §1	2/3 to 2/7
Wed 2/5	j-invariants, singular Weierstrass cubics	[S1] ch. III §1-2
Fri 2/7	the algebraic group law	[S1] ch. III §3
Mon 2/10	isogenies	[S1] ch. III §4	2/10 to 2/17
Wed 2/12	invariant differential	[S1] ch. III §5
Fri 2/14	dual isogeny	[S1] ch. III §6
Mon 2/17	dual isogeny, applications to torsion	[S1] ch. III §6
Wed 2/19	Tate module, Galois representations	[S1] ch. III §7	2/19
Fri 2/21	The Weil pairing	[S1] ch. III §8	2/21 to 2/26
Mon 2/24	Weil pairing on Tate module, applications	[S1] ch. III §8
Wed 2/26	Endomorphisms	[S1] ch. III §9-10
Fri 2/28	Endomorphisms in positive characteristic	[S1] ch. V §3	2/28 and 3/5
Mon 3/3	Elliptic curves over \(\mathbb{C}\)	[S1] ch. VI, [S2] ch. I	recording, notes
Wed 3/5	Hasse bound, Weil conjectures	[S1] ch. V §1-2
Fri 3/7	Over local fields: reduction and torsion	[S1] ch. VII §1-3	3/7 to 3/10
Mon 3/10	Action of inertia, reduction types	[S1] ch. VII §4-5
Wed 3/12	Weak Mordell–Weil pt 1	[S1] ch. VIII §1	3/12 to 3/17
Fri 3/14	no class
Mon 3/17	Weak Mordell–Weil pt 2, Descent Theorem	[S1] ch. VIII §1, 3
Wed 3/19	Mordell’s Theorem, heights on \(\mathbb{P}^N\)	[S1] ch. VIII §4, 5	3/19 to 3/21
Fri 3/21	More heights	[S1] ch. VIII §5
Spring Break
Mon 3/31	Heights on elliptic curves	[S1] ch. VIII §6	3/31 to 4/2
Wed 4/2	Finishing MW!	[S1] ch. VIII §6
Fri 4/4	Canonical Height	[S1] ch. VIII §8	4/4
Mon 4/7	Diophantine Approximation	[S1] ch. VIII §1,2	4/7 to 4/9, Silverman’s comments on \(\nu\)-adic distance on curve
Wed 4/9	\(\nu\)-adic distance on a curve	[S1] ch. VIII §2
Fri 4/11	applications of Siegel’s Theorem	[S1] ch. VIII §2-3	4/11 to 4/14
Mon 4/14	proof of Siegel’s Theorem	[S1] ch. VIII §3
Wed 4/16	\(S\)-unit equations	[S1] ch. VIII §4	4/16
Fri 4/18	Kummer via Galois cohomology	[S1] ch. X §1	4/18-4/21
Mon 4/21	Computing the MW Group, \(2\)-descent	[S1] ch. X §1	4/21
Wed 4/23	Twists of curves	[S1] ch. X §2	4/25 and 4/30
Fri 4/25	Presentation – Chirag
Mon 4/30	Principal homogenous spaces	[S1] ch. X §3
Wed 4/28	Presentation – Vibhu
Fri 5/2	Selmer and Shafarevich–Tate groups	[S1] ch. X §4	5/2
Thurs 5/8	Presentations – Tanis, André, Julian