Identity component

Abstract: Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the Galois group of each variational equation of arbitrary order along any integral curve This was conjectured by the first author based on a suggestion by the third author The first-order non-integrability criterion, obtained by the first and second authors using only first variational equations, is extended to higher orders by the present criterion Using this result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped the first order criterion

Abstract: We let G be the isometry group of a nondegenerate symmetric or skew-symmetric form h on a space V over some local or global field K [i.e., G = U(V, h)]. We let LG be the L group of Go = the identity component of G and r be the natural representation of this group. We develop here a theory of L functions and e factors attached to this r. In particular we develop a theory of local and global zeta integrals for such G. We use very special intertwining operators; therein we require an analysis of the proper normalization of such operators. With such techniques we determine the correct e factors associated to local representations. The issue of the correct local L factor is discussed.

Abstract: A new bound is obtained for the function g ( h ), whose existence was proved by Ledermann & Neumann (1956), such that p h divides the order of the automorphism group of a finite group G , if p g ( h ) divides the order of G .

Abstract: In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on \(S^5 \sharp k(S^2 \times S^3)\) for each positive integer k.