N-Covers of hyperelliptic curves

N-covers of Hyperelliptic Curves

For a hyperelliptic curve C of genus g with a divisor class of order n = g + 1, we shall consider an associated covering collection of curves Dδ , each of genus g2. We describe, up to isogeny, the Jacobian of each Dδ via a map from Dδ to C, and two independent maps from Dδ to a curve of genus g(g − 1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number f...

Arithmetic of Hyperelliptic Curves

Automorphisms of Hyperelliptic Modular Curves X0(n) in Positive Characteristic

We study the automorphism groups of the reduction X0(N)× F̄p of a modular curve X0(N) over primes p ∤ N .

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 ...

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...