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**Fields of definition of abelian varieties with real multiplication.**
*(English)*
Zbl 0877.14030

Childress, Nancy (ed.) et al., Arithmetic geometry. Conference on arithmetic geometry with an emphasis on Iwasawa theory, March 15-18, 1993, Arizona State Univ., Tempe, AZ, USA. Providence, RI: American Mathematical Society. Contemp. Math. 174, 107-118 (1994).

Let \(K\) be a field, \(\overline{K}\) its algebraic closure, and \(A\) an abelian variety defined over \(\overline{K}\). Assume that \(A\) satisfies the following two conditions: (1) \(\text{End}_{\overline{K}} (A)\otimes\mathbb{Q}\) is a totally real number field \(F\) of degree \(d=\dim A\) over \(\mathbb{Q}\); (2) for each \(g\in G=\text{Gal} (\overline{K}/K)\) there exists an \(F\)-equivariant isogeny \(\mu_g:{}^gA\to A\) defined over \(\overline{K}\), where \({}^gA\) is the twist of \(A\) with respect to \(g\). Such an \(A\) with \(K=\mathbb{Q}\) can arise, e.g., as a \(\overline{\mathbb{Q}}\)-factor up to isogeny of an abelian variety \(C\) over \(\mathbb{Q}\) whose algebra of endomorphisms defined over \(\mathbb{Q}\) is a field of degree \(\dim C\) over \(\mathbb{Q}\) after tensoring with \(\mathbb{Q}\). (It is proved in the present paper that \(C\) is actually a power of \(A\) over \(\overline{\mathbb{Q}}\) up to isogeny.) The main theorem of the paper is that such an \(A\) is \(F\)-equivariantly isogenous over \(\overline{K}\) to an abelian variety \(B\) with \(\text{End}_{\overline{K}} (B)\otimes\mathbb{Q}=F\) defined over a finite \((2,\dots,2)\)-extension of \(k\).

The proof is divided into five parts: (1) One can find \(B\) over a finite extension \(E\) of \(K\) if and only if the class \(\gamma\in H^2(G,F^*)\) defined by the cocycle \(\{c(g,h)= \mu_g\circ {}^g\mu_h\circ \mu_{gh}^{-1}\}\) vanishes after the restriction to \(H^2(\text{Gal} (\overline{K}/E), F^*)\). (This is a rather standard argument.)

(2) \(\gamma\in H^2(G,F^*)[2]\), i.e. \(\gamma\) is a 2-torsion. (Fixing a polarization on \(A\), one can define the degree of an isogeny as an element of \(F^*\). Since \(F\) is totally real, the Rosati involution is trivial which implies that \(c(g,h)^2=\deg c(g,h)=\deg \mu_g\cdot\deg \mu_h/\deg \mu_{gh}\).)

(3) \(H^2(G,F^*)[2]\cong H^2(G,\{\pm1\})\oplus H^2(G,P)[2]\), where \(P\) is defined by the exact sequence \(1\to\{\pm1\}\to F^*\to P\to 1\). (This follows from the fact that \(P\) is free, \(\pm1\) being the only torsion in \(F^*\).)

(4) \(H^2(G,P)[2]\cong \operatorname{Hom}(G,P/P^2)\) (since \(P\) is free).

(5) Let \(\gamma= \gamma_\pm+ \overline{\gamma}\) \((\gamma_\pm\in H^2(G, \{\pm1\})\), \(\overline{\gamma}\in \operatorname{Hom}(G,P/P^2))\) be the decomposition of \(\gamma\). \(\overline{\gamma}\) is trivialized by the extension \(K_P\) of \(K\) corresponding to its kernel, which is evidently of type \((2,\dots,2)\). As for \(\gamma_\pm\), it is known that the group \(H^2(G,\{\pm1\})\) is 0 if \(\text{char }K=2\). In the case of \(\text{char }K\neq 2\), the author employs a theorem of Merkurev that \(\text{Br}(K)[2]\cong (G,\{\pm1\})\) is generated by the classes of quaternion algebras over \(K\).

An interesting question is wheter \(\gamma\) becomes zero over \(K_P\), because it does if \(d=1\) (Elkies). The author proves this assumption under two additional hypotheses: \(\text{char } K=0\) and \(d\) is odd. According to him, the former should be superfluous and it is an interesting question whether one can or cannot drop the latter.

For the entire collection see [Zbl 0802.00017].

The proof is divided into five parts: (1) One can find \(B\) over a finite extension \(E\) of \(K\) if and only if the class \(\gamma\in H^2(G,F^*)\) defined by the cocycle \(\{c(g,h)= \mu_g\circ {}^g\mu_h\circ \mu_{gh}^{-1}\}\) vanishes after the restriction to \(H^2(\text{Gal} (\overline{K}/E), F^*)\). (This is a rather standard argument.)

(2) \(\gamma\in H^2(G,F^*)[2]\), i.e. \(\gamma\) is a 2-torsion. (Fixing a polarization on \(A\), one can define the degree of an isogeny as an element of \(F^*\). Since \(F\) is totally real, the Rosati involution is trivial which implies that \(c(g,h)^2=\deg c(g,h)=\deg \mu_g\cdot\deg \mu_h/\deg \mu_{gh}\).)

(3) \(H^2(G,F^*)[2]\cong H^2(G,\{\pm1\})\oplus H^2(G,P)[2]\), where \(P\) is defined by the exact sequence \(1\to\{\pm1\}\to F^*\to P\to 1\). (This follows from the fact that \(P\) is free, \(\pm1\) being the only torsion in \(F^*\).)

(4) \(H^2(G,P)[2]\cong \operatorname{Hom}(G,P/P^2)\) (since \(P\) is free).

(5) Let \(\gamma= \gamma_\pm+ \overline{\gamma}\) \((\gamma_\pm\in H^2(G, \{\pm1\})\), \(\overline{\gamma}\in \operatorname{Hom}(G,P/P^2))\) be the decomposition of \(\gamma\). \(\overline{\gamma}\) is trivialized by the extension \(K_P\) of \(K\) corresponding to its kernel, which is evidently of type \((2,\dots,2)\). As for \(\gamma_\pm\), it is known that the group \(H^2(G,\{\pm1\})\) is 0 if \(\text{char }K=2\). In the case of \(\text{char }K\neq 2\), the author employs a theorem of Merkurev that \(\text{Br}(K)[2]\cong (G,\{\pm1\})\) is generated by the classes of quaternion algebras over \(K\).

An interesting question is wheter \(\gamma\) becomes zero over \(K_P\), because it does if \(d=1\) (Elkies). The author proves this assumption under two additional hypotheses: \(\text{char } K=0\) and \(d\) is odd. According to him, the former should be superfluous and it is an interesting question whether one can or cannot drop the latter.

For the entire collection see [Zbl 0802.00017].

Reviewer: T.Ooe (MR 95i:11057)