We give a non trivial upper bound for the number of elliptic curves Er,s : Y 2 = X3 + rX + s with (r, s) ∈ [R + 1, R + M ]× [S + 1, S + M ] that are isomorphic to a given curve. We also give an almost optimal lower bound for the number of distinct isomorphic classes represented by elliptic curves Er,s with the coefficients r, s lying in a small box.

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.

Note that Γ(1) = Γ1(1) = Γ0(1) = SL2(Z). We now define the modular curves X(N) = H∗/Γ(N), X1(N) = H/Γ1(N), X0(N) = H/Γ0(N). and similarly define Y (N), Y1(N), and Y0(N), with H∗ replaced by H. Following the same strategy we used for X(1), one can show that these are all compact Riemann surfaces. Having defined the modular curves X(N), X1(N), and X0(N), we now want to consider the meromorphic fu...

is a pseudo-effective divisor on Y . We have an application to a problem concerning the positivity of divisors on the moduli space of stable curves, namely, let Mg (resp. Mg) be the moduli space of stable (resp. smooth) curves of genus g ≥ 2. Let λ be the Hodge class and δi’s (i = 0, . . . , [g/2]) are boundary classes. Then, a divisor (8g+4)λ− gδ0− ∑[g/2] i=1 4i(g− i)δi is pseudo-effective, an...

A Pythagorean–hodograph (PH) curve r(t) = (x(t), y(t), z(t)) has the distinctive property that the components of its derivative r(t) satisfy x(t)+y(t)+z(t) = σ(t) for some polynomial σ(t). Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x(t), y(t), z(t) in terms of polynomials u(t), v(t), p(t), ...

Consider the smooth projective models C of curves y = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g →∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves, via Chabauty’s me...

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra’s ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery’...

The magnetization measurements at 5 K were carried out for Ni2Mn1 − xCuxGa (0 ≤ x ≤ 0.40) and Ni2MnGa1 − yCuy (0 ≤ y ≤ 0.25) alloys. All of the magnetization curves are characteristic of ferromagnetism or ferrimagnetism. By using Arrott plot analysis the spontaneous magnetization of all samples was determined from the magnetization curves. The magnetic moment per formula unit, μs, at 5 K was es...

Gauss’s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form E(λ) : y = x(x− 1)(x− λ). We study a modular function which “measures” the variation of periods for the isomorphic curves E(λ) and E ( λ λ−1 ) , and we show that it padically “interpolates” the cusp form for the “congruent number” curve E(2), the case where these pairs...

The Griiths formalism is applied to nd constant torsion curves which are extremal for arclength with respect to variations preserving torsion, xing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions. In the calculus of variations under nonholonomic constraints, one tries to nd the...