Elliptic loops

Given a local ring (R,m) and an elliptic curve E(R/m), we define its elliptic loop as the points of P2(R) projecting to E under the canonical modulo-m reduction, endowed with an operation that extends the curve's addition. While its subset of points satisfying the curve's Weierstrass equation is a group, this larger object is proved to be a power-associative abelian algebraic loop, which is seldom completely associative. When an elliptic loop has no points of order 3, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over R, which we call layers. Stronger associativity properties are established when me vanishes for small values of e∈Z. When the underlying ring is R=Z/peZ, the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description.