 Tags:

Resources: #MOC/resources
 https://crypto.stanford.edu/pbc/notes/elliptic/weil2.html #resources/fullcourses
 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 #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/coursenotes

MIT OCW Course #resources/fullcourses #resources/coursenotes

Links:
 Hasse bound
 curve
 Hasse bounds
 Weierstrass equation
 Weierstrass p function
 theta function
 sigma function
 Jacobian
 padic height
 padic 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
 NeronOggShaferevich
elliptic curve
Notes
Definitions
 Definition: an elliptic curve is a smooth projective (modules) genus 1 curve with a rational point.
Supersingular

Definition: an elliptic curve is supersingular iff the associated formal group has height 2, or equivalently \(E[p^n](\mkern 1.5mu\overline{\mkern1.5muk\mkern1.5mu}\mkern 1.5mu) = 1\) 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.

\(E\) 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