, On the other hand, comparing Proposition 7.9 and the growth condition (a) of (i) shows that f (X, w) must be reducible in F [X, w, Suppose that (i) holds. Since F is a subfield of Q[X]/f (X, w 0 ) which is of degree d over Q, we see that e divides d

, Let K be a CM-field of degree 2g, let ? 0 (w) ? K[w], r 0 (w) ? Q[w] and let k 2 be an integer. We say that (? 0 , r 0 ) represents a family of g-dimensional varieties with embedding degree k and complex multiplication by K if: (i)

, ii) there is an infinite set W 0 of integers such that p 0 (w 0 ) and r 0 (w 0 ) are prime numbers for all w 0 ? W 0

N. , 1) and ? k (p 0 (w)) are both divisible by r 0 (w)

, Assuming a weak form of the Bateman-Horn-Conrad heuristics, this definition is essentially equivalent to Definition 3.6 in Freeman

, (w)) and suppose that C w (X) is irreducible. If w 0 ? W 0 , then C w0 (X) is the characteristic polynomial of the corresponding abelian variety, and we obtain a polynomial family in the sense of Definition 7.1 with fixed CM-field K. Conversely, let P be a polynomial family in the sense of Definition

, the roots of C w0 (X) are p 0 (w 0 )-Weil numbers

. Thus, Freeman's definition is essentially equivalent to our Definition 7.1 of families P having fixed CM-field K. Finally

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