Tuples of hyperelliptic curves y2=xn+a

TUPLES OF HYPERELLIPTIC CURVES y² = x^n + a

In this paper it is shown that, given n ∈ Z ≥3 and a, b, c ∈ Q × , there exists a polynomial d(t) ∈ Q[t] such that the curves over Q(t) given by y 2 = x n + ad(t) resp. y 2 = x n + bd(t) and y 2 = x n + cd(t) all have a Jacobian with positive Mordell-Weil rank over Q(t). Extensions of this result to sets of four curves are discussed, as well as the problem of demanding in addition that d(t) is ...

Arithmetic of Hyperelliptic Curves

Families of Hyperelliptic Curves

Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...

Hyperelliptic Curves and Cryptography

In 1989, Koblitz proposed using the jacobian of a hyperelliptic curve defined over a finite field to implement discrete logarithm cryptographic protocols. This paper provides an overview of algorithms for performing the group law (which are necessary for the efficient implementation of the protocols), and algorithms for solving the hyperelliptic curve discrete logarithm problem (whose intractab...

Pointless hyperelliptic curves

In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus.