Small zeros of quadratic forms over number fields. II

Abstract

Let F F be a nontrivial quadratic form in N N variables with coefficients in a number field k k and let Z \mathcal {Z} be a subspace of k N {k^N} of dimension M , 1 ≤ M ≤ N M,1 \leq M \leq N . If F F restricted to Z \mathcal {Z} vanishes on a subspace of dimension L , 1 ≤ L > M L,1 \leq L > M , and if the rank of F F restricted to Z \mathcal {Z} is greater than M − L M - L , then we show that F F must vanish on M − L + 1 M - L + 1 distinct subspaces X 0 , X 1 , … , X M − L {\mathcal {X}_0},{\mathcal {X}_1}, \ldots ,{\mathcal {X}_{M - L}} in Z \mathcal {Z} each of which has dimension L L . Moreover, we show that for each pair X 0 , X 1 , 1 ≤ l ≤ M − L {\mathcal {X}_0},{\mathcal {X}_1},1 \leq l \leq M - L , the product of their heights H ( X 0 ) H ( X