Title: Daniel's Flashcards / Year 4 / Term 2 / Elliptic Curves

html: Definition. Pic ⁡ ( E ) = Div ⁡ ( E ) ∼, Pic ⁡ 0 ( E ) = Div ⁡ 0 ( E ) ∼. where Div ⁡ 0 ( E ) = { D ∈ Div ⁡ E | deg ⁡ ⁡ D = 0 }.

html: Definition. Hom ⁡ ( E 1 , E 2 ) = { isogenies E 1 → E 2 } ∪ { 0 }.

html: Definition. For ϕ , ψ ∈ End ⁡ ( E ) = Hom ⁡ ( E , E ), we put ⟨ ϕ , ψ ⟩ = deg ⁡ ( ϕ + ψ ) − deg ⁡ ϕ − deg ⁡ ψ and tr ⁡ ( ϕ ) = ⟨ ϕ , 1 ⟩.

html: Definition. The reduction Ẽ Ẽ ∕ k of E ∕ K is defined by the reduction of a minimal Weierstrass equation.
E has good reduction if Ẽ Ẽ is non-singular (and so an elliptic curve) otherwise has bad reduction.
For an integral Weierstrass equation,
- If v ( Δ ) = 0 then good reduction.
- If 0 < v ( Δ ) < 1 2 then bad reduction.
- If v ( Δ ) ≥ 1 2 then beware that the equation might not be minimal.