Abstract:
This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.

Abstract:
This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply.

Abstract:
In this paper, building on prior joint work by Mallios and Ntumba, we show that $\mathcal A$-\textit{transvections} and \textit{singular symplectic }$\mathcal A$-\textit{automorphisms} of symplectic $\mathcal A$-modules of finite rank have properties similar to the ones enjoyed by their classical counterparts. The characterization of singular symplectic $\mathcal A$-automorphisms of symplectic $\mathcal A$-modules of finite rank is grounded on a newly introduced class of pairings of $\mathcal A$-modules: the \textit{orthogonally convenient pairings.} We also show that, given a symplectic $\mathcal A$-module $\mathcal E$ of finite rank, with $\mathcal A$ a \textit{PID-algebra sheaf}, any injective $\mathcal A$-morphism of a \textit{Lagrangian sub-$\mathcal A$-module} $\mathcal F$ of $\mathcal E$ into $\mathcal E$ may be extended to an $\mathcal A$-symplectomorphism of $\mathcal E$ such that its restriction on $\mathcal F$ equals the identity of $\mathcal F$. This result also holds in the more general case whereby the underlying free $\mathcal A$-module $\mathcal E$ is equipped with two symplectic $\mathcal A$-structures $\omega_0$ and $\omega_1$, but with $\mathcal F$ being Lagrangian with respect to both $\omega_0$ and $\omega_1$. The latter is the analog of the classical \textit{Witt's theorem} for symplectic $\mathcal A$-modules of finite rank.

Abstract:
Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, the number of Gamma_i(F_2)-orbits in F_2^m is equal to 3\times 2^s, where s is the number of simple reflections that appear in a reduced decomposition for u or v and F_2 is the two-element field.

Abstract:
Symmetric symplectic spaces of Ricci type are a class of symmetric symplectic spaces which can be entirely described by reduction of certain quadratic Hamiltonian systems in a symplectic vector space. We determine, in a large number of cases, if such a space admits a subgroup of its transvection group acting simply transitively. We observe that the simply transitive subgroups obtained are one dimensional extensions of the Heisenberg group.

Abstract:
It is shown that the existence of an $\omega$-compatible Einstein metric on a compact symplectic manifold $(M,\omega)$ imposes certain restrictions on the symplectic Chern numbers. Examples of symplectic manifolds which do not satisfy these restrictions are given. The results offer partial support to a conjecture of Goldberg.

Abstract:
We prove a reciprocity law for one-dimensional compatible systems of mod p representations of absolute Galois groups of number fields. We prove that these arise from Hecke characters, and in particular recover by purely algebraic means the classical result (that is usually proven using deep results of Waldschmidt in transcendental number theory) that one-dimensional compatible p-adic systems arise from Hecke characters.

Abstract:
In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.

Abstract:
In this paper we study the topology of the space $\I_\omega$ of complex structures compatible with a fixed symplectic form $\omega$, using the framework of Donaldson. By comparing our analysis of the space $\I_\omega$ with results of McDuff on the space $\cat J_\omega$ of compatible almost complex structures on rational ruled surfaces, we find that $\I_\omega$ is contractible in this case. We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff.

Abstract:
The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section of the paper analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer will require developments in transcendence theory in characteristic $p$.