Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that they form an abelian group.
翻译:椭圆曲线是数字理论和代数几何学中的基本对象,在一字段上的点根据几何加法形成一个边缘群。 任何字段上的椭圆曲线都承认一个Weierstrass模型,但先前正式证明,该添加法与该模型有关联的要么是先进的代数几何法,要么是重复的计算,特别是在特征二中。我们在Lean 理论证明中正式确定,一个Weierstrasurs曲线的非星系点在任何特性的字段上的类型,并且纯粹的代数证明,它们构成一个贝利亚群。