6,298 research outputs found
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 be an abelian scheme over a geometrically connected variety
defined over , a finitely generated field over . Let be
the generic point of and a closed point. If and
are the Lie algebras of the -adic Galois
representations for abelian varieties and , then
is embedded in by specialization. We
prove that the set closed point is independent of and confirm Conjecture 5.5 in [2].Comment: 4 page
Adelic openness without the Mumford-Tate conjecture
Let be a non-singular projective variety over a number field , a
non-negative integer, and V_{\A}, the etale cohomology of 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 in H(\A_f)
under the adelic Galois representation \rho_{\A}: G_K ->
\Aut(V_{\A})=\GL_n(\A_f), where is the Hodge group. The motivating example
is a celebrated theorem of Serre, which asserts that if is an elliptic
curve without complex multiplication over and , 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
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