Elliptic curve arithmetic and superintegrable systems

Abstract

Harmonic oscillator and the Kepler problem are superintegrable systems which admit more integrals of motion than degrees of freedom and all these integrals are polynomials in momenta. We present superintegrable deformations of the oscillator and the Kepler problem with algebraic and rational first integrals. Also, we discuss a family of superintegrable metrics on the two-dimensional sphere, which have similar first integrals.The prime divisors are points on X, denoted P i = (x i , y i ), including point at infinity P ∞ , which plays the role of neutral element 0 in arithmetic of elliptic curves.

New superintegrable models on spaces of constant curvature

“…Similarly, in the fourth quadrant of βα plane i.e. for α ∈ (−1, 0) and β = α+1 we obtain the functioñ f (ϕ) ≡f IV (ϕ) = (α + 1)(cos 2νϕ − α) − 2 √ α sin 2νϕ | cos νϕ | 1 + α 2 − 2α cos 2νϕ (43) and the related potential…”

Superintegrable systems and Riemann-Roch theorem

“…we can "restore" the symmetry between points by using substitution So, all the superintegrable systems described in Section 2 remain superintegrable after the noncanonical transformations (3.22), see discussion in [20,22,23].…”

The Kepler Problem: Polynomial Algebra of Nonpolynomial First Integrals

“…Let us consider non-canonical transformations of momenta preserving symmetries of configuration space, but breaking symmetry between divisors [16,36,37,38,39,41,42]. It is easy to see, that transformation of momenta…”

“…According to [41,42] these Hamiltonians (2.11) are superintegrable Hamiltonians because this noncanonical transformation sends the original sum of elliptic integrals (2.5) to the sum…”

“…At m = n all first integrals x 3 , y 3 (2.14) and C mn (2.15) are algebraic functions in phase space. Some explicit expressions of these first integrals may be found in [42] . Now we are ready to formulate the main result in this note.…”