Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13

Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems

2019-02-19T23:23:32Z

Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems Kawakami, H. This is the last part of a series of three papers entitled ''Four-dimensional Painlevé-type equations associated with ramified linear equations''. In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlevé-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system.

Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space

2019-02-19T23:23:29Z

Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space Zhang, J.; Hu, N. We realize the Hopf algebra Uq(sp₂n) as an algebra of quantum differential operators on the quantum symplectic space X(fs;R) and prove that X(fs;R) is a Uq(sp₂n)-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of Uq(sp₂n).

The Chazy XII Equation and Schwarz Triangle Functions

2019-02-19T23:24:14Z

The Chazy XII Equation and Schwarz Triangle Functions Bihun, O.; Chakravarty, S. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation y′′′−2yy′′+3y′²=K(6y′−y²)², K∈C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y=a₁w₁+a₂w₂+a₃w₃ where wi solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients (a1,a2,a3) and the Darboux-Halphen parameters (α,β,γ), which are enumerated in Table 2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α,β,γ;z) which are used to represent the solutions wi of the Darboux-Halphen system. The paper only considers the case where α+β+γ<1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P^,Q^,R^).

Twists of Elliptic Curves

2019-02-19T23:23:45Z

Twists of Elliptic Curves Kronberg, M.; Soomro, M.A.; Top, J. In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set H¹(GK¯/K,AutK¯(E)). The results are illustrated by examples.