elliptic curve

elliptic curve

Notes

Definitions

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  • 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 \(E[p^n](\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu) = 1\) 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.

  • \(E\) is supersingular if and only if its endomorphism algebra (over {\overline {K))) is an order in a quaternion algebra

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Classification

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Ranks

Moduli

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Uniformization

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L functions

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Analytic rank

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Issues with representability

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Torsion

Mazur’s theorem: attachments/Pasted%20image%2020220323094427.pngattachments/Pasted%20image%2020220323163947.png

Galois representations

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Fermat

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