Torsion of Abelian Varieties, Weil Classes and Cyclotomic Extensions

Let K ⊂ C be a field finitely generated over Q, K(a) ⊂ C the algebraic closure of K, G(K) = Gal(K(a)/K its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime l we write K(l) for the subfield of K(a) obtained by adjoining to K all l−power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab) ⊂ K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K = Q then Q(ab) = Q(c) (the Kronecker-Weber theorem). We write χl : G(K) → Z ∗ l for the cyclotomic character defining the Galois action on all l-power roots of unity. We write χ = χl mod l : G(K) → Z ∗ l → (Z/lZ) ∗ for the cyclotomic character defining the Galois action on the l−th roots of unity. Let g a positive integer, X a g−dimensional abelian variety over K. We write End(X) for the ring of all endomorphisms of X defined over K and End(X) for the finite-dimensional semisimple Q−algebra End(X)⊗Q. Its center F = FX is a field if and only if X is K−isogenous to a power of a K−simple abelian variety. If so, F is either a totally real number field or a CM-field. We write Lie(X) for the tangent space to X at the origin. It is the g−dimensional K−vector space, which carries a natural structure of faithful End(X)⊗Q K−module. The well-knownMordell-Weil-Néron-Lang theorem asserts thatX(K) is a finitely generated commutative group. In particular, its torsion subgroup TORS(X(K)) is finite. Hereafter we will write TORS(A) for the torsion subgroup of a commutative group A. This implies that TORS(X(L)) is finite for any finite algebraic extension L of K. Mazur [7] has raised the question of whether the groups X(K(l)) are finitely generated. In this connection, Serre (in letters to Mazur, of January 1974) and Imai [6] have proved independently that TORS(X(K(l))) is finite for all l. Now assume that L ⊂ K(a) ⊂ C is a possibly infinite Galois extension of K. When L = K(c) a theorem of Ribet [10] asserts that TORS(X(K(c)) is finite. The author [20] has proven that if the center F of End(X) is a direct sum of totally real number fields and TORS(X(L)) is infinite then L contains infinitely many roots of unity. On the other hand, Bogomolov (Séminaire Delange-Pisot-Poitou, mai 1982, Paris) proved that TORS(X(L)) is finite if the intersection of L and K(ab) has finite degree over K. For example, if K = Q, we obtain that if TORS(X(L)) is infinite then the intersection of L and Q(c) has infinite degree over Q. The main result of the present paper is the following statement, which deals with essentially non-cyclotomic extensions and may be viewed as a partial improvement of the Bogomolov’s result.