On an analogue of the Lutz–Nagell Theorem for hyperelliptic curves

We produce a version of the Lutz–Nagell Theorem for hyperelliptic curves of genus g⩾1. We consider curves C defined by y2=f(x), where f is a monic polynomial of degree 2g+1 defined over the ring of integers of a number field F or its non-archimedean completions. If J is the Jacobian of C, and ϕ is the Abel–Jacobi map of C into J sending the point at ∞ of our model of C to the origin of J, we show that if P=(a,b) is a rational point of C such that ϕ(P) is torsion in J, then a and b are integral if the order n of P is not a prime power, and bound the denominators of a and b if n is. When a and b are integral, we give criteria for when b2 divides the discriminant Δ(f) of f. Finally we show for f∈Z[x], that if P=(a,b)∈C(Q) and ϕ(P) is a torsion point of order n⩾2, then we have a,b∈Z, and either b=0 or b2 divides Δ(f).