Paper 2025/736

Superglue: Fast formulae for $(2,2)$ gluing isogenies

Abstract

We study the structure of theta structure on products of elliptic curves, detailing their construction through the symmetries induced by $4$-torsion points. In particular, we show how these symmetries allow the computation of theta structures projectively, thus avoiding the use of modular inversions. Furthermore, we explore the self-similarity of the matrix representation of theta structures, arising from the action of the canonical $2$-torsion point in the Kummer line. Combined with the sparsity of some $4$-torsion points, this structure leads to new formulae for computing gluing $(2,2)$ isogenies that require significantly fewer precomputations and arithmetic operations. These new equations also naturally support the evaluation of points on the quadratic twist at a negligible additional cost, without requiring operations in a field extension.