Linear equivalence between elliptic curves in Weierstrass and Hesse form

Elliptic curves in Hesse form admit more suitable arithmetic than ones in Weierstrass form. But elliptic curve cryptosystems usually use Weierstrass form. It is known that both those forms are birationally equivalent. Birational equivalence is relatively hard to compute. We prove that elliptic curves in Hesse form and in Weierstrass form are linearly equivalent over initial field or its small extension and this equivalence is easy to compute. If cardinality of finite field q = 2 (mod 3) and Frobenius trace T = 0 (mod 3), then equivalence is defined over initial finite field. This linear equivalence allows multiplying of an elliptic curve point in Weierstrass form by passing to Hessian curve, computing product point for this curve and passing back. This speeds up the rate of point multiplication about 1,37 times

Hesse Pencils and 3-Torsion Structures

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three

Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time

We present families of (hyper)elliptic curve which admit an efficient deterministic encoding function

The topology of Calabi-Yau threefolds

We ask about the simply connected compact smooth 6-manifolds which can support structures of Calabi-Yau threefolds. In particular, we study the interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We have a cup-product cubic form on the second integral cohomology, a linear form given by the second Chern class, and the integral middle cohomology, and if the homology is torsion free this information determines precisely the diffeomorphism class of the underlying 6-manifold by a result of Wall. For simplicity, we assume that the cubic form defines a smooth real elliptic curve whose Hessian is also smooth. Under a further relatively mild assumption that there are no non-movable surfaces $E$ on $X$ with $1 \le E^3 \le 8$, we prove that the real elliptic curve must have two connected components rather than one, and that the K\"ahler cone is contained in the open positive cone on the bounded component; we show moreover that the second Chern class is also positive on this open cone. Using Wall's result, for any given third Betti number we therefore have an abundance of examples of smooth compact oriented 6-manifolds which support no Calabi-Yau structures, both in the cases when the cubic defines a real elliptic curve with one or two connected components.Comment: 31 pages, 6 figures. Minor improvements from v2, significant changes between v1 and v2. arXiv admin note: text overlap with arXiv:2011.1287