An effective proof of the hyperelliptic Shafarevich conjecture

2013/10/24 by von Känel, Rafael
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1310.6727

Abstract

Let C be a hyperelliptic curve of genus g≥ 1 over a number field K with good reduction outside a finite set of places S of K. We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g, S and K. In particular, we obtain that for any given number field K, finite set of places S of K and integer g≥ 1 one can in principle determine the set of K-isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside S.

Related

• Siegel's Theorem and the Shafarevich Conjecture
• Fiber products of hyperelliptic curves
• Potential good reduction of hyperelliptic curves
• Global Weierstrass equations of hyperelliptic curves
• Counting hyperelliptic curves that admit a Koblitz model