- Tags:
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Resources: #MOC/resources
- https://crypto.stanford.edu/pbc/notes/elliptic/weil2.html #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
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Haiyang’s notes:
- attachments/EllipticCurve notes1.pdf #resources/notes
- attachments/EllipticCurve notes2.pdf #resources/notes
- attachments/EllipticCurve notes3.pdf #resources/notes
- Course notes: https://www.math.columbia.edu/~phlee/CourseNotes/EllipticCurves.pdf#page=1 #resources/course-notes
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MIT OCW Course #resources/full-courses #resources/course-notes
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Links:
- Hasse bound
- curve
- Hasse bounds
- Weierstrass equation
- Weierstrass p function
- theta function
- sigma function
- Jacobian
- p-adic height
- p-adic uniformization
- moduli stack of elliptic curves
- j invariant
- conductor
- hyperellptic involution
- Unsorted/rational points
- level of an elliptic curve
- weight of an elliptic curve
- conductor of an elliptic curve
- discrete log problem
- abelian variety
- Problems:
- Unsorted/nodal curve
- Unsorted/potentially semistable
- Unsorted/twist
- Neron-Ogg-Shaferevich
elliptic curve
Notes
Definitions
- Definition: an elliptic curve is a smooth projective (modules) genus 1 curve with a rational point.
Supersingular
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Definition: an elliptic curve is supersingular iff the associated formal group has height 2, or equivalently \(E[p^n](\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu) = 1\) for all \(k\) (so trivial group of geometric points of order \(p\).)
- Idea: unusually large endomorpism algebras, e.g. an order in a quaternion algebra.
-
\(E\) is ordinary iff not supersingular.
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\(E\) is supersingular if and only if its endomorphism algebra (over
) is an order in a quaternion algebra
Classification
Ranks
Moduli
Uniformization
L functions
Analytic rank
Issues with representability
Torsion
Mazur’s theorem: 
Galois representations
Fermat
