On the a b c $abc$ conjecture in algebraic number fields

Abstract

In this paper, we prove a weak form of the $abc$ conjecture generalised to algebraic number fields. Given integers satisfying $a+b=c$, Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height $H_K(a,b,c)$ in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers $a,b,c$ in a number field K satisfying $a+b=c$, we give an upper bound for the logarithm of the projective height $H_L(a,b,c)$ in terms of norms of prime ideals dividing $abc{O}_L$, where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical $G:=G(a,b,c)$ as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S-unit equations, our estimates imply the upper bound