Deficient points on extensions of abelian varieties by Gm

AbstractLet K be a number field, and take x ∈ K∗. It is an elementary fact that x is a root of unity if and only if x satisfies the following condition: for almost all primes l, x is an lth power in the field obtained by adjoining the lth roots of unity to K. The analogous statement need no longer be true if K∗ is replaced by the group G(K), where G is an extension by Gm of an abelian variety over K. For a given G, we determine all x which satisfy the condition; these are the “deficient points” in our title