RATIONAL EQUIVARIANT FORMS

We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.

On the Moduli of a Quantized Elastica in ℙ and KdV Flows: Study of Hyperelliptic Curves as an Extension of Euler's Perspective of Elastica I

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli [Formula: see text] of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space [Formula: see text]. Using the topology, [Formula: see text] is classified. Studies on a loop space in the category of topological spaces Top are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry DGeom, we also proved an existence of a functor between a triangle category related to a loop space in Top and that in DGeom using the induced topology. As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on [Formula: see text], this paper devotes relations between hyperelliptic curves and a quantized elastica on ℙ as an extension of Euler's perspective of elastica.