Genus 2 curves with bad reduction at one odd prime

Abstract

In this article we consider smooth projective curves C of genus two described by integral equations of the form y 2 = xh(x), where h(x) ∈ Z[x] is monic of degree 4. It turns out that if h(x) is reducible, then the absolute discriminant of C can never be an odd prime, except when h(x) = (x − b)g(x) and g(x) is irreducible. In this case we obtain a complete description of such genus 2 curves. In fact, we prove that there are four one-parameter families C i t , i = 1, 2, 3, 4, of such curves such that if C is a genus two curve with an odd prime absolute discriminant, then C is C i t , for some i, 1 ≤ i ≤ 4, and t ∈ Z. Moreover, we show that C i t has an odd prime absolute discriminant, p, if and only if a certain degree-4 irreducible polynomial f i (t) ∈ Z[t] takes the value p at t. Hence there are conjecturally infinitely many such curves. When h(x) is irreducible, we give explicit examples of one-parameter families of genus 2 curves Ct such that Ct has an odd prime absolute discriminant for conjecturally infinitely many integer values t.