Torsion of abelian varieties, Weil classes and cyclotomic extensions

Abstract

Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μ_m) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-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 χ_{[lscr ]}[ratio ]G(K)→Z*_{[lscr ]} for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ_{[lscr ]}= χ_{[lscr ]} mod [lscr ][ratio ]G(K) →Z*_{[lscr ]}→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ_{[lscr ]} identifies Gal (K([lscr ])/K) with a subgroup of Z*_{[lscr ]}=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*_{[lscr ]}. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z_{[lscr ]})-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z_{[lscr ]}) and Gal (Q([lscr ])′/Q)= Z*_{[lscr ]}/μ(Z_{[lscr ]}) is isomorphic to Z_{[lscr ]}.