Research proof index

The classical formula for the quadratic twist away from characteristic 2. Suppose char K ≠ 2, so that after completing the square we may assume E has the form y² = x³ + a₂x² + a₄x + a₆, and suppose L = K(α) where α² = d is a nonsquare in K (every separable quadratic extension arises this way when char K ≠ 2). Then the quadratic twist of E...

Twisting the Galois action. The Galois action on the points of the quadratic twist is the Galois action on the points of E, twisted by the quadratic character of L/K: for σ ∈ Aut(M/K), transporting the action of σ on Eᴸ(M) through Eᴸ(M) ≅ E(M) gives χ(σ) times the action of σ on E(M). Taking M to be a separable closure of K, this...

Naturality of quadraticTwistPointEquiv in M: the isomorphisms on M-points over varying M ⊇ L are all induced by a single isomorphism of curves over L, so they commute with the maps on points induced by any L-algebra homomorphism.

The anti-equivariance underlying quadraticTwistPointEquiv_galois: if σ ∈ Aut(M/K) does not fix L pointwise (χ(σ) = -1), then transporting its action through Eᴸ(M) ≅ E(M) gives minus its action. This is the point-level shadow of quadraticTwistVarChange_baseChange_map.

The M-level form of the twist's defining cocycle: any σ ∈ Aut(M/K) not fixing L pointwise (i.e. with χ(σ) = -1) conjugates the base change to M of quadraticTwistVarChange by the automorphism [-1] of E. This is the base change of quadraticTwistVarChange_map.

The defining cocycle of the quadratic twist. The nontrivial σ ∈ Gal(L/K) conjugates the change of variables quadraticTwistVarChange (carrying E to Eᴸ) by the automorphism [-1] of E. This is the datum expressing that Eᴸ is the descent of E along the twisted Galois action σ ↦ (-1) ∘ σ, and it holds for every j.