Integral points on punctured abelian varieties

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