We calculate explicitly the j-invariants of the elliptic curves corresponding to rational points on the modular curve X+ ns(11) by giving an expression defined over Q of the j-function in terms of the function field generators X and Y of the elliptic curve X+ ns(11). As a result we exhibit infinitely many elliptic curves over Q with nonsplit mod 11 representations.

A new elliptic curve scalar multiplication algorithm is proposed. The algorithm o ers about twice the troughput of some conventional OEF-base algorithms because it combines the Frobenius map with the table reference method based on baseexpansion. Furthermore, since this algorithm suits conventional computational units such as 16, 32 and 64 bits, its base eld F p m is expected to enhance ellipti...

In order to prove that semistable elliptic curves E over Q are modular, A. Wiles [Wiles 95] considers the Galois representation ρ : Gal(Q/Q) −→ GL2(F3) provided by the 3-torsion points E[3] of a semistable elliptic curve E. He proves that certain rather restricted deformations of ρ are modular. It follows then that, in particular, the deformation provided by the Galois representation ρE : Gal(Q...

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions ...

In this paper we consider the problem of calculating the higherorder Weierstrass weight of the branch points of a superelliptic curve C. For any q > 1, we give an exact formula for the q-weight of an affine branch point. We also find a formula for the q-weight of a point at infinity in the case where n and d are relatively prime. With these formulas, for any fixed n, we obtain an asymptotic for...

Let n be an integer such that the modular curve X0(n) is hyperelliptic of genus ≥ 2 and such that the Jacobian of X0(n) has rank 0 over Q. We determine all points of X0(n) defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to Q-isomorphism, all but finitely many elliptic curves with n-isogenies over quadratic fields are in fact ...

The “Q-curves” Q1(c) = c,Q2(c) = c + c, . . . , Qn(c) = (Qn−1(c)) + c = f c (0) have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg’s map fc(x) = x + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x + c as well — by “bifurcati...

Let’s understand first what means “straight” line in the plane. If you want to go from a point in a plane “straight” to another one, your trajectory will be such a line. In other words, a straight line L has the property that if we fix two points P and Q on it, then the piece of L between P and Q is the shortest curve in the plane which joins the two points. Now, if instead of a plane (“flat” s...

We prove that distortion of a knotted curve in R is great than 4.76. This improves a result obtained by John M. Sullivan and Elizabeth Denne in [DS]. 1 The definition and results M. Gromov introduced the notion of distortion for submanifolds of R in [G], p.113-114. In the case of a simple closed curve K (closed 1-dimensional submanifold of R) distortion is U(K) = max P,Q∈K |PQ|K |PQ| , where |P...

We give a detailed account of the use of Q-curve reductions to construct elliptic curves over Fp2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of ...