Singular reduction of Nambu–Poisson manifolds

The version of Marsden–Ratiu Poisson reduction theorem for Nambu–Poisson manifolds by a regular foliation have been studied by Ibáñez et al. In this paper, we show that this reduction procedure can be extended to the singular case. Under a suitable notion of Hamiltonian flow on the reduced space, we show that a set of Hamiltonians on a Nambu–Poisson manifold can also be reduced.

Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/ℚ and non-torsion point P ∈ E(ℚ), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequence v(Ψn) of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on ĥ(P)/h(E) for integer points having two large integral multiples.

QUANTIZATION OF SINGULAR REDUCTION

This paper creates a theory of quantization of singularly reduced systems. We compare our results with those obtained by quantizing algebraically reduced systems. In the case of a Kähler polarization, we show that quantization of a singularly reduced system commutes with reduction, thus generalizing results of Sternberg and Guillemin. We illustrate our theory by treating an example of Arms, Gotay and Jennings where algebraic and singular reduction at the zero level of the momentum mapping differ. In spite of this, their quantizations agree.