On ℓ-independency in families of motivic ℓ-adic representations

Abstract

Let k be a finitely generated field of characteristic 0 embedded into \({\mathbb{C},\,X}\) a smooth, separated and geometrically connected scheme over k with generic point \({\eta}\) and \({f : Y \rightarrow X}\) a smooth proper morphism. Let \({f^{an}_{\mathbb{C}} : Y^{an}_{\mathbb{C}} \rightarrow X^{an}_{\mathbb{C}}}\) denote the associated morphism of complex analytic spaces. For \({x \in X(\mathbb{C})}\), write H for the Betti cohomology of \({Y^{an}_{\mathbb{C}, x}}\) with coefficients in \({\mathbb{Q}}\) and for \({x \in X}\), write \({{\rm H}_{\ell}}\) for the l-adic cohomology of \({Y_{\overline{x}}}\) with coefficients in \({\mathbb{Q}_{\ell}}\) (Under our assumptions on \({f : Y \rightarrow X,\,{\rm H}}\) and \({{\rm H}_{\ell}}\) are independent of x). For every prime l, let \({X^{ex}_{\ell}}\) be the set of all \({x \in X}\) where the Zariski closure \({G_{\ell, x}}\) of the image of the Galois representation \({\Gamma_{k(x)} \rightarrow {\rm GL}({\rm H}_{\ell})}\) has dimension strictly smaller than the dimension of \({G_{\ell,\eta}}\). By previous works of A. Tamagawa and the author, \({X^{ex}_{\ell}}\) is ‘small’ in the sense that if X is a curve then for every integer \({\delta \geq 1}\) the set of all \({x \in X^{ex}_{\ell}}\) with \({[k(x) : k] \leq \delta}\) is finite. Set \({X^{ex} := {\bigcap}_{\ell} X^{ex}_{\ell}}\). The Tate conjectures predict that for every \({x \in X}\) the \({G_{\ell, x}}\) are defined over \({\mathbb{Q}}\), reductive and independent of l hence, in particular, that the sets \({X^{ex}_{\ell}}\) are independent of l. Let \({\overline{G}}\) denote the Zariski closure of the image of the monodromy representation \({\pi_{1} (X_{\mathbb{C}}^{an}; \, x) \rightarrow {\rm GL}({\rm H})}\). Then \({\overline{G}}\) is a semi-simple algebraic group of rank—say—r. The main result of this note is that for \({x \notin X^{ex},\,G_{\ell, x} \cap \overline{G}_{\mathbb{Q}_{\ell}}}\) is a semi-simple algebraic group of rank r. This implies in particular that: (1) If \({\overline{G}_{\overline{\mathbb{Q}}}}\) has only simple factors of type \({A_{n}}\) then \({X^{ex}_{\ell}}\) is independent of l; (2) For every prime l and \({x \notin X^{ex}}\) the unipotent radical of \({G_{\ell, x}}\) coincides with the unipotent radical of \({G_{\ell, \eta}}\) and, in particular, is independent of \({x \notin X^{ex}}\); (3) For every prime l, if there exists \({x_{\ell} \in X}\) such that \({G_{\ell, x_{\ell}}}\) is reductive then for every \({x \notin X^{ex},\,G_{\ell, x}}\) is reductive. (3) applies in particular when \({{\rm H}}\) is a geometrically irreducible \({\overline{G}}\)-module. This implies, for instance, that apart from a few exceptional cases, for every r-tuple \({\underline{d}=(d_{1}, \dots, d_{r})}\) of integers \({\geq 2}\) there exists a non-singular complete intersection in \({\mathbb{P}^{n+r}_{\mathbb{Q}}}\) with multi-degree \({\underline{d}}\) for which the Tate semi-simplicity conjecture holds (for every prime l).