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

Abstract

We produce a version of the Lutz–Nagell Theorem for hyperelliptic curves of genus g ⩾ 1 . We consider curves C defined by y 2 = f ( x ) , where f is a monic polynomial of degree 2 g + 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 b 2 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 b 2 divides Δ ( f ) .