2020/03/19 by Andrzej Dąbrowski, Dabrowski, Andrzej, Mohammad Sadek +1
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2003.09010
openalex publication_date 2020/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we consider smooth projective curves C of genus two described by integral equations of the form y2=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 two one-parameter families Cti, i=1,2, of such curves such that if C is a genus two curve with an odd prime absolute discriminant, then C is Cti, for some i, and t∈ℤ. Moreover, we show that Cti has an odd prime absolute discriminant, p, if and only if a certain degree-4 irreducible polynomial fi(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.