Abelian varieties having a reduction of \(K\)3 type
Let \(X\) be an abelian variety defined over a number field \(K\) and let \(K(a)\) be an algebraic closure of \(K\) and \(G(K)=\text{Gal}(K(a)/K)\) its Galois group. Let \(T_ \ell(X)\) be the Tate module of \(X\) \((\ell\) being a prime number) and \(\rho_ \ell:G(K) \to \text{Aut} T_ \ell(X) \subset \text{Aut} V_ \ell(X)\) the \(\ell\)-adic representation of \(G(K)\), where \(V_ \ell(X)=T_ \ell \otimes_{\mathbb{Z}_ \ell}\mathbb{Q}_ \ell\). The author defines the Lie algebra \({\mathcal G}_ \ell\) as the Lie algebra of \(\text{Im} \rho_ \ell \subset \text{Aut} V_ \ell(x)\). Let \(E\) be the \(\mathbb{Q}\)-algebra \(\text{End}(X) \otimes_ \mathbb{Z} \mathbb{Q}\) and \(E(\ell)=E \otimes_ \mathbb{Q} \mathbb{Q}_ \ell\), one has \(E(\ell) \subset \text{End} V_ \ell(X)\) and, by a theorem of \textit{G. Faltings} [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)], \(E(\ell)=\text{End}_{{\mathcal G}(\ell)}V_ \ell(X)\). Given an invertible sheaf \({\mathcal L}\) on \(X_ \mathbb{C}\) its Riemann form \(H\) induces a skew-symmetric bilinear form: \({\mathcal H}_ \ell:V_ \ell(X)\times V_ \ell(X) \to \mathbb{Q}_ \ell\). Let Sp\(_ \ell\) be the Lie algebra of the symplectic group of \(V_ \ell(X)\) defined by \({\mathcal H}_ \ell\). The main result in this paper is the following: Let \(X\) be an abelian variety over \(K\) verifying that there exists a nonarchimedean place \(v\) of \(K\) such that \(X\) has a good reduction at \(v\) and this reduction is a simple abelian variety, \(X(v)\), over \(K(v)\) of \(K3\)-type, then: (a) The \(\mathbb{Q}\)-algebra \(\text{End}(X) \otimes_{\mathbb{Z}} \mathbb{Q}\) is a commutative field. (b) For all prime numbers \(\ell\): \({\mathcal G}_ \ell \cap\text{Sp}_ \ell=\text{Sp}_ \ell \cap \text{End}_{E(\ell)}V_ \ell(X)\).
scientific article; zbMATH DE number 62993
scientific article