THEOREM 1. Let K/Q be a number field, and let C/K be a smooth projective curve of genus at least 2. For each^class ^ e i / 1 (Gal (AT/AT), Aut(C)), let C% be the twist of C by x (cf. [11, X §3]) and le\Jx = J ac (Q be the Jacobian variety of Cx. There is a constant y = y{C/K) such that #C,( /Q^r7 a n k * ( K ) forallxeH{G2\{K/K),A\xi{C)). We illustrate this general theorem by applying it to a p...

In this work, we consider the number of integer solutions of Diophantine equation D : y − 2yx − 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep : y = Pp(x) = y p 1 + y p 2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x− and y−coordinates of all rational points (x, y) on Ep over Fp. ...

‎It is shown that the knowledge of a surjective morphism $Xto Y$ of complex‎ ‎curves can be effectively used‎ ‎to make explicit calculations‎. ‎The method is demonstrated‎ ‎by the calculation of $j(ntau)$ (for some small $n$) in terms of $j(tau)$ for the elliptic curve ‎with period lattice $(1,tau)$‎, ‎the period matrix for the Jacobian of a family of genus-$2$ curves‎ ‎complementing the classi...

An elliptic curve y = x +ax+ b over the prime field Fp may be twisted by a change of variables into the curve y = x + a′x + b′, where a′ differs from a by a quartic residue. We show that most curves over Fp may be twisted to have a′ quite small, allowing one of the multiplications in the point doubling formula to be replaced by additions. This speeds up algorithms to solve the elliptic curve di...

A hybrid BEM-p-y curves approach was developed for the single pile analysis with free/fixed head restraint conditions. The method considers the soil non-linear behaviour by means of p-y curves in series to a multi-layered elastic half-space. The non-linearity of reinforced concrete pile sections, also considering the influence of tension-stiffening, has been considered. The model reproduces the...

Let P (x, y) be a rational polynomial. If the curve (P (x, y) = k), k ∈ Q, is irreducible and admits an infinite number of points whose coordinates are integers, Siegel’s theorem implies that the curve is rational. We deal with the case where k is a generic value and prove, in the spirit of the Abhyankar-MohSuzuki theorem, that there exists an algebraic automorphism sending P (x, y) to the poly...

We consider the question of characterizing the behavior of parametric curves whose components are cubic polynomials. When there is no chance of confusion, we will refer to such curves as cubic curves with the understanding that each of x(t) and y(t) are themselves cubic polynomials. We classify various types of parametric cubics using their defining coefficients. We show that this can be done i...

The use of elliptic and hyperelliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve. Recently, Satoh proposed a probabilistic polynomial time algorithm to test whether the Jacobian – over a finite field Fq – of a hyperelliptic curve of the form Y 2 = X + aX + bX (with a, b ∈ Fq) has a large prime factor. His approach is to obtain candidates for the ...

K Kedlaya proposed an method to count the number of Fq rational points in a hyper elliptic curve using the Leschetz xed points formula in Monsky Washinitzer Cohomology The method has been extended to super elliptic curves Gaudry and G urel immediately to characteristic two hyper elliptic curves and to Cab curves J Denef F Vercauteren Based on Miura theory which is associated with how a curve is...

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing 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 ...