On the rationality of algebraic monodromy groups of compatible systems

Let E be a number field and X be a smooth geometrically connected variety defined over a characteristic p finite field F_q. Given an n-dimensional pure E-compatible system of semisimple \lambda-adic representations \rho_\lambda of the fundamental group \pi_1(X) with connected algebraic monodromy groups G_\lambda, we construct a common E-form G of all the groups G_\lambda. In the absolutely irreducible case, we construct a common E-form i:G->GL_{n,E} of all the tautological representations G_\lambda->GL_{n,E_\lambda} and a G-valued adelic representation \rho_A^G of \pi_1(X) such that their composition is isomorphic to the product representation of all \rho_\lambda. Moreover, if X is a curve and the (absolute) outer automorphism group of G^der is trivial, then the \lambda-components of \rho_A^G form an E-compatible system of G-representations. Analogous rationality results in characteristic zero, predicted by the Mumford-Tate conjecture, are obtained under some conditions including ordinariness.Comment: 35 pages. Thm. 1.1(ii) is improved so that G sits in GL_{n,E

Invariant dimensions and maximality of geometric monodromy action

Let X be a smooth separated geometrically connected variety over F_q and f:Y-> X a smooth projective morphism. We compare the invariant dimensions of the l-adic representation V_l and the F_l-representation \bar V_l of the geometric \'etale fundamental group of X arising from the sheaves R^wf_*Q_l and R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a maximality result of the geometric monodromy action on V_l whenever \bar V_l is semisimple and l is sufficiently large. We also provide examples for \bar V_l to be semisimple for l>>0

Specialization of monodromy group and l-independence

Let $E$ be an abelian scheme over a geometrically connected variety $X$ defined over $k$, a finitely generated field over $\mathbb{Q}$. Let $\eta$ be the generic point of $X$ and $x\in X$ a closed point. If $\mathfrak{g}_l$ and $(\mathfrak{g}_l)_x$ are the Lie algebras of the $l$-adic Galois representations for abelian varieties $E_{\eta}$ and $E_x$, then $(\mathfrak{g}_l)_x$ is embedded in $\mathfrak{g}_l$ by specialization. We prove that the set $\{x\in X$ closed point $| (\mathfrak{g}_l)_x\subsetneq \mathfrak{g}_l\}$ is independent of $l$ and confirm Conjecture 5.5 in [2].Comment: 4 page

Adelic openness without the Mumford-Tate conjecture

Let $X$ be a non-singular projective variety over a number field $K$, $i$ a non-negative integer, and V_{\A}, the etale cohomology of $\bar X$ with coefficients in the ring of finite adeles \A_f over \Q. Assuming the Mumford-Tate conjecture, we formulate a conjecture (Conjecture 1.2) describing the largeness of the image of the absolute Galois group $G_K$ in H(\A_f) under the adelic Galois representation \rho_{\A}: G_K -> \Aut(V_{\A})=\GL_n(\A_f), where $H$ is the Hodge group. The motivating example is a celebrated theorem of Serre, which asserts that if $X$ is an elliptic curve without complex multiplication over $\bar K$ and $i=1$, then \rho_{\A}(G_K) is an open subgroup of \GL_2(\hat \Z)\subset \GL_2(\A_f). We state and in some cases prove a weaker conjecture which does not require Mumford-Tate but which, together with Mumford-Tate, implies Conjecture 1.2. We also relate our conjectures to Serre's conjectures on maximal motives.Comment: Section 5 is ne