Abelian varieties without homotheties (Q2372680)

Let \(X\) be an abelian variety defined over a field \(K\) and \(\ell\) a prime number different from the characteristic of \(K\). Let \(G_K\) be the absolute Galois group of \(K\), \(T_{\ell}(X)\) the \(\ell\)-adic Tate module of \(X\) and \(V_{\ell}(X):=T_{\ell}(X)\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}\). There is a natural Galois representation \(\rho_{\ell,X}:G_K\to\text{Aut}_{\mathbb{Z}_{\ell}}(T_{\ell}(X))\subset\text{Aut}_{\mathbb{Q}_{\ell}}(V_{\ell}(X))\). The image \(G_{\ell,X}\) of \(\rho_{\ell,X}\) is a compact \(\ell\)-adic Lie subgroup of \(\text{Aut}_{\mathbb{Q}_{\ell}}(V_{\ell}(X))\). Denote by \(\mathfrak{g}_{\ell,X}\) its Lie algebra. Again this is viewed as a Lie \(\mathbb{Q}_{\ell}\)-subalgebra in \(\text{End}_{\mathbb{Q}_{\ell}}(V_{\ell}(X))\). In the particular case where \(K\) is a global field of characteristic \(p>2\), the author had previously obtained the following results [Math. Notes 22, 493--498 (1978); translation from Mat. Zametki 22, 3--11 (1977; Zbl 0419.14021)]: \(\mathfrak{g}_{\ell,X}\) is a reductive \(\mathbb{Q}_{\ell}\)-algebra decomposed as \(\mathfrak{g}_{\ell,X}\cong\mathfrak{g}^{\text{ss}}\oplus\mathfrak{c}\), where \(\mathfrak{g}^{\text{ss}}\) is a semisimple \(\mathbb{Q}_{\ell}\)-algebra and \(\mathfrak{c}\) is the center of \(\mathfrak{g}_{\ell,X}\); \(\dim_{\mathbb{Q}_{\ell}}(\mathfrak{c})=1\); and if \(\mathfrak{C}(X)\) (which is the center of \(\text{End}(X)\otimes\mathbb{Q}\)) is a product of totally real number fields, then \(\mathfrak{c}=\mathbb{Q}_{\ell}\cdot\text{id}\). \textit{F. A. Bogomolov} [C. R. Acad. Sci. Paris Sér. A 290, 701--703 (1980; Zbl 0457.14020); Izv. Akad. Nauk SSSR, Ser. Mat. 44, 782--804 (1980; Zbl 0453.14018)] proved that when \(K\) is a number field, then \(\mathfrak{g}_{\ell,X}\) always contains the homotheties \(\mathbb{Q}_{\ell}\cdot\text{id}\). However, when \(K\) is a global field of characteristic \(p\), this does not hold. The counter-example is given by an ordinary elliptic curve \(Z\) defined over a finite field \(k\) which is extended to \(Z\times_kK\). In the paper under review the author gives another instance where \(\mathfrak{g}_{\ell,X}\) does not contain the homotheties. More precisely, it is produced an absolutely simple abelian variety \(X\) defined over a global field \(K\) of characteristic \(p>2\) which is not isogenous over the algebraic closure of \(K\) to an abelian variety over a finite field and is such that \(\mathfrak{g}_{\ell,X}\cap(\mathbb{Q}_{\ell}\cdot\text{id})=\{0\}\).