On the image of Galois \(l\)-adic representations for abelian varieties of type III
Let \(F\) be a number field with absolute Galois group \(G_{F}\). Let \(A\) be a simple abelian variety defined over \(F\). If (i) \(\mathcal{R}:=\text{End}_{\bar{F} }(A)=\)End\(_{F}(A)\), (ii) \(\mathcal{R} \otimes_{\mathbb{Z}}\mathbb{\mathbb{Q}}\) is of type III, (iii) \(G_{l} ^{\text{alg}}\), the Zariski closure of the image of the \(l\)-adic Galois representation \(\rho_{l}\) in \(\text{GL}_{V_{l}(A)}/\mathbb{Q}_{l}\), is a connected algebraic group, and (iv)\ \(\dim A=2h[E:\mathbb{Q}]\) where \(h\) is odd and \(E\) is the center of \(\mathcal{R}\otimes_{\mathbb{Z}}\mathbb{\mathbb{Q}}\); then \(A\) is said to be of class \(\mathcal{B}\). The work under review is a study of the image of \(\rho_{l}\) when \(A\) is an abelian variety of class \(\mathcal{B}\). After an embedding \(F\hookrightarrow\mathbb{C}\) we can define a cocharacter \(\mu_{\infty}:\mathbb{G}_{m}(\mathbb{C})\to \text{GL}_{2g}(\mathbb{C})\). The smallest algebraic subgroup of \(\text{GL}_{2g}\) defined over \(\mathbb{Q}\) such that its geometric points include the image of \(\mu_{\infty}\) is called the Mumford-Tate group -- its base change to \(\mathbb{Q}_{l}\) is denoted MT\((A)_{\mathbb{Q}_{l}}\). One always has \((G_{l}^{\text{alg}})^{\circ}\subset\)MT\((A)_{\mathbb{Q} _{l}}\) and a well-known conjecture of Mumford and Tate states that these two groups are equal. Here, it is shown that the Mumford-Tate conjecture holds whenever \(A\) is of class \(\mathcal{B}\); more precisely we have \(G_{l}^{\text{alg}}=\text{MT}(A)_{\mathbb{Q}_{l}}\) in this case since the first group is connected by assumption. It is a conjecture of Lang that for large \(l\) the group \(\rho_{l}(G_{F})\) contains all of the homotheties in \(\text{GL}_{T_{l}(A)}(\mathbb{Z}_{l})\). This is proven in the case that \(A\) is of class \(\mathcal{B}\), in fact it follows immediately from the result above. For any profinite group \(G\), let \(G^{\prime}\) be the closure of the commutator subgroup of \(G\). Then for large \(l\) we have \(\rho_{l}(G_{F} ^{\prime\prime})\) is a product of groups of the form \(\text{SO}_{(T_{\lambda},\psi_{\lambda})}(\mathcal{O}_{\lambda})^{\prime},\) where \(\lambda\mid l\) and SO\(_{(T_{\lambda},\psi_{\lambda })}\) is the connected component of the identity in the group of all \(X\in\text{GL}_{T_{\lambda}}\) such that \(\psi_{\lambda}(Xv_{1},Xv_{2})=\psi_{\lambda}(v_{1},v_{2})\) for all \(v_{1},v_{2}\in T_{\lambda}\). Similarly, for large \(l\) the image of \(G_{F}^{\prime\prime}\) under the representation \(\bar{\rho}_{l}:G_{F} \to\text{GL}(A\left[ l\right] )\) is a product of \(\text{SO}_{(A_{\lambda}[\lambda],\bar{\psi} _{k})}(k_{\lambda})^{\prime}\). As a consequence, if \(\Lambda\) is the Riemann lattice such that \(A(\mathbb{C})=\mathbb{C}^{g}/\Lambda\), and \(\kappa\) is the \(\mathbb{Z}\)-bilinear pairing \(\Lambda\times\Lambda\to\mathbb{Z}\) given by the polarization of \(A\), and \(\mathcal{C}_{\mathcal{R}}(\text{Sp}_{(\Lambda ,\kappa)})\) is the centralizer of \(\mathcal{R}\) in Sp\(_{(\Lambda,\kappa)}\), then \(\mathcal{C}_{\mathcal{R}}(\text{Sp}_{(\Lambda,\kappa)})\) decomposes as a product of \(\text{SO}_{(T_{\lambda},\psi_{\lambda})}\) for \(l>>0\). Finally, the open image theorem of Serre is generalized. It is shown that \(\rho_{l}(G_{F})\) is open in \(\mathcal{C}_{\mathcal{R}}(\text{Sp}_{(\Lambda,\kappa)})(\mathbb{Z} _{l})\) for all \(l\). Furthermore, if \(r(l)\) is the number of primes in \(\mathcal{O}_{E}\) lying over \(l\) then the index of \(\rho_{l}(G_{F}^{\prime})\) in \(\mathcal{C}_{\mathcal{R}}(\text{Sp}_{(\Lambda,\kappa)})(\mathbb{Z} _{l})\) divides \(2^{r(l)}\) and \(\rho_{l}(G_{F}^{\prime\prime})=\mathcal{C}_{\mathcal{R}}(\text{Sp} _{(\Lambda,\kappa)})(\mathbb{Z}_{l}^{\prime })\)for \(l \gg 0\).
scientific article; zbMATH DE number 5780337
scientific article