--- date: 2021-10-11 17:52 modification date: Tuesday 19th October 2021 21:59:49 title: elliptic curve aliases: [elliptic curve, elliptic curves, CM] --- - Tags: - #AG/elliptic-curves - Resources: #MOC/resources - #resources/full-courses - The Arithmetic of Elliptic Curves by Joseph Silverman #resources/books - Rational points on elliptic curves, UTM, Springer, by Joseph Silverman and John Tate #resources/books - Haiyang's notes: - [attachments/EllipticCurve notes1.pdf](attachments/EllipticCurve%20notes1.pdf) #resources/notes - [attachments/EllipticCurve notes2.pdf](attachments/EllipticCurve%20notes2.pdf) #resources/notes - [attachments/EllipticCurve notes3.pdf](attachments/EllipticCurve%20notes3.pdf) #resources/notes - Course notes: #resources/course-notes - MIT OCW Course #resources/full-courses #resources/course-notes - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes2.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes3.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes4.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes5.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes6.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes7.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes8.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes9.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes10.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes11.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes12.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes13.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes14.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes15.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes16.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes17.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes18.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes19.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes20.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes21.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes22.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes23.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes24.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes25.pdf) - [ ] [**notes**](https://math.mit.edu/classes/18.783/2017/LectureNotes26.pdf) - Links: - [Hasse bound](Hasse%20bound.md) - [curve](Unsorted/curves.md) - [Hasse bounds](Hasse%20bounds.md) - [Weierstrass equation](Weierstrass%20equation) - [Weierstrass p function](Weierstrass%20p%20function.md) - [theta function](theta%20function.md) - [sigma function](sigma%20function) - [Jacobian](Jacobian.md) - [p-adic height](p-adic%20height) - [p-adic uniformization](p-adic%20uniformization) - [moduli stack of elliptic curves](moduli%20stack%20of%20elliptic%20curves.md) - [j invariant](j%20invariant.md) - [conductor](conductor.md) - [hyperellptic involution](hyperellptic%20involution) - [Unsorted/rational points](Unsorted/rational%20points.md) - [level of an elliptic curve](level%20of%20an%20elliptic%20curve) - [weight of an elliptic curve](weight%20of%20an%20elliptic%20curve) - [conductor of an elliptic curve](conductor%20of%20an%20elliptic%20curve) - [discrete log problem](discrete%20log%20problem.md) - [abelian variety](Unsorted/abelian%20variety.md) - Problems: - [Sato-Tate conjecture](Unsorted/Sato-Tate%20conjecture.md) - [Fermat's Last Theorem](Unsorted/Fermat's%20Last%20Theorem.md) - [BSD](Unsorted/Birch%20and%20Swinnerton-Dyer%20conjecture.md) - [Unsorted/nodal curve](Unsorted/nodal%20curve.md) - [Unsorted/potentially semistable](Unsorted/potentially%20semistable.md) - [Unsorted/twist](Unsorted/twist.md) - [Neron-Ogg-Shaferevich](Neron-Ogg-Shaferevich) # elliptic curve # Notes ## Definitions ![](attachments/Pasted%20image%2020220430214853.png) - Definition: an **elliptic curve** is a [smooth](smooth) [projective (modules)](Unsorted/projective%20(modules).md) [genus](genus%20of%20a%20curve) 1 curve with a [rational point](rational%20point). - Embedding into $\PP^2\slice\CC$: ![2023-01-09-4](attachments/2023-01-09-4.png) ## Supersingular - Definition: an elliptic curve is **supersingular** iff the associated formal group has height 2, or equivalently $E[p^n](\bar k) = 1$ for all $k$ (so trivial group of geometric points of order $p$.) - Idea: unusually large [endomorpism algebras](endomorpism%20algebras.md), e.g. an [order](order.md) in a quaternion algebra. - $E$ is **ordinary** iff not supersingular. - $E$ is supersingular if and only if its endomorphism algebra (over ![{\overline {K))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d5ec352e79f883030b85cbfb6600ad03b72663)) is an [order](order.md) in a [quaternion algebra](quaternion%20algebra.md) ![](attachments/Pasted%20image%2020220408193619.png) ![](attachments/Pasted%20image%2020220516192012.png) ## Classification ![attachments/Pasted%20image%2020211005011249.png](attachments/Pasted%20image%2020211005011249.png) ## Ranks ![Pasted%20image%2020211106013928.png](Pasted%20image%2020211106013928.png) ## Moduli ![](attachments/Pasted%20image%2020220220024735.png) ## CM Generating the [Hilbert class field](Unsorted/Hilbert%20class%20field.md): ![2023-01-09-5](attachments/2023-01-09-6.png) See generalization to [abelian varieties](Unsorted/abelian%20variety.md) # Uniformization ![](attachments/Pasted%20image%2020220204094135.png) ![](attachments/Pasted%20image%2020220204094431.png) # L functions ![](attachments/Pasted%20image%2020220217213717.png) ![](attachments/Pasted%20image%2020220430214930.png) ## Analytic rank ![](attachments/Pasted%20image%2020220408193957.png) # Issues with representability ![](attachments/Pasted%20image%2020220220024927.png) ![](attachments/Pasted%20image%2020220220024916.png) ![](attachments/Pasted%20image%2020220220030853.png) ![](attachments/Pasted%20image%2020220220032224.png) # Torsion Mazur's theorem: ![](attachments/Pasted%20image%2020220323094427.png)![](attachments/Pasted%20image%2020220323163947.png) # Galois representations ![](attachments/Pasted%20image%2020220323164020.png) ![](attachments/Pasted%20image%2020220323164043.png) ![](attachments/Pasted%20image%2020220323164106.png) # Fermat ![](attachments/Pasted%20image%2020220430215025.png)