Research proof index

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite continuous R₀-algebras R i with the discrete topology whose cardinalites are unifomly bounded. Then 𝒰(Rᵢ) ≃ₐ[R] R i for F-many i.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous ring homs f i : R →+* R i, there exists F-many i such that 𝒰(Rᵢ) ≃+* R i and this map is compatible with f.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous surjective ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also surjective.

The rational points of the quadratic twist, viewed inside E(L) via the isomorphism over L, are exactly the points of E(L) on which the nontrivial element of Gal(L/K) acts as -1 (just as E(K) consists of the points on which it acts as +1). One inclusion is Affine.Point.map_baseChange (the base change of a K-point is σ-fixed) together wi...

An explicit L-isomorphism (Eᶿ)ᴸ ≅ Eᴸ (the change of variables of the module docstring) which moreover is anti-equivariant for the Galois action: its conjugate by the nontrivial σ ∈ Gal(L/K) differs from it by the automorphism [-1] of E. This nontrivial cocycle is the origin of the twist being a nontrivial form of E.

Classification of the forms of E split by L/K, for j(E) ∉ {0, 1728}: an elliptic curve over K which becomes isomorphic to E over L is isomorphic over K either to E or to its quadratic twist by L. (Such forms are classified by H¹(Gal(L/K), Aut(E_L)) = Hom(ℤ/2, {±1}), which has order 2. For j ∈ {0, 1728} the automorphism group is large...