Integral points on punctured abelian varieties

Abstract

Let A/{mathbb {Q}} be an abelian variety such that A({mathbb {Q}}) ={0_A}. Let ell and p be rational primes, such that A has good reduction at p, and satisfying ell equiv 1 ,(mathrm{mod} ,p) and ell not mid # ,A({mathbb {F}}_p). Let S be a finite set of rational primes. We show that (A setminus {0_A})({mathscr {O}}_{L,S}) =varnothing for 100% of cyclic degree ell fields L/{mathbb {Q}}, when ordered by conductor, or by absolute discriminant.