Rational Orbits Research Articles

Graded Lie algebras, compactified Jacobians and arithmetic statistics

A simply laced Dynkin diagram gives rise to a family of curves over \mathbb{Q} and a coregular representation, using deformations of simple singularities and Vinberg theory, respectively. Thorne conjectured and partially proved a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne’s picture and show that 2-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck–Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives.

Arithmetic properties of 3-cycles of quadratic maps over [formula omitted

It is shown that c=−29/16 is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map fc(x)=x2+c has a rational periodic point of period 3. Several arithmetic conditions on the set of all such rational numbers c and the rational orbits of fc(x) are proved. A graph on the numerators of the rational 3-periodic points of maps fc is considered which reflects connections between solutions of norm equations from the cubic field of discriminant −23.

Bound orbits around modified Hayward black holes

The neutral time-like particle’s bound orbits around modified Hayward black holes have been investigated. We find that both in the marginally bound orbits (MBO) and the innermost stable circular orbits (ISCO), the test particle’s radius and its angular momentum are all more sensitive to one of the parameters [Formula: see text]. Especially, modified Hayward black holes with [Formula: see text] could mimic the same ISCO radius around the Kerr black hole with the spin parameter up to [Formula: see text]. Small [Formula: see text] could mimic the ISCO of small-spinning test particles around Schwarzschild black holes. Meanwhile, rational (periodic) orbits around modified Hayward black holes have also been studied. The epicyclic frequencies of the quasi-circular motion around modified Hayward black holes are calculated and discussed with respect to the observed Quasi-periodic oscillations (QPOs) frequencies. Our results show that rational orbits around modified Hayward black holes have different values of the energy from the ones of Schwarzschild black holes. The epicyclic frequencies in modified Hayward black holes have different frequencies from Schwarzschild and Kerr ones. These might provide hints for distinguishing modified Hayward black holes from Schwarzschild and Kerr ones by using the dynamics of time-like particles around the strong gravitational field.

On the topology of geometric and rational orbits for algebraic group actions over valued fields, II

The aim of this paper is twofold. First, we show that if G is a smooth nilpotent group acting on an algebraic variety V defined over an admissible valued field k and then the Zariski closedness of the geometric orbit in is equivalent to the Hausdorff closedness of the rational orbit in V(k). Second, we provide some calculations for the fact that there is a bijection between the set of G(k)-orbits and the kernel of the natural map in flat cohomology. These results are obtained in the framework of studying the rational orbits.

On the topology of geometric and rational orbits for algebraic group actions over valued fields

Abstract In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types D 4 {D_{4}} , E 6 {E_{6}} , E 7 {E_{7}} , E 8 {E_{8}} , G 2 {G_{2}} on an affine variety defined over a local function field k, and v ∈ V ⁢ ( k ) {v\in V(k)} such that the geometric orbit G . v {G.v} is Zariski closed although the corresponding relative orbit G ⁢ ( k ) . v {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

Rational orbits of primitive trivectors in dimension six

Let $G=\operatorname{GL}(1)\times \mathrm{GSp}(6)$ and $V$ be the irreducible representation of $G$ of dimension 14 over a field of characteristic not equal to 2,3. This is an irreducible prehomogeneous vector space. We determine generic rational orbits and their stabilizers of this prehomogeneous vector space.

Rational orbits of the space of pairs of exceptional Jordan algebras

Let k be a field of characteristic not equal to 2,3, O an octonion over k and J the exceptional Jordan algebra defined by O. We consider the prehomogeneous vector space (G,V) where G=GE6×GL(2) and V=J⊕J. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with k-isomorphism classes of pairs (M,n) where M's are isotopes of J and n's are cubic étale subalgebras of M. Also we prove that if O is split, then generic rational orbits are in bijective correspondence with isomorphism classes of separable extensions of k of degrees up to 3.

Arithmetic exponents in piecewise-affine planar maps

We consider the growth of some indicators of arithmetical complexity of rational orbits of (piecewise) affine maps of the plane, with rational parameters. The exponential growth rates are expressed by a set of exponents; one exponent describes the growth rate of the so-called logarithmic height of the points of an orbit, while the others describe the growth rate of the size of such points, measured with respect to the p-adic metric. Here p is any prime number which divides the parameters of the map. We show that almost all the points in a domain of linearity (such as an elliptic island in an area-preserving map) have the same set of exponents. We also show that the convergence of the p-adic exponents may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behaviour of these quantities in the chaotic regions, in both area-preserving and dissipative systems. In the former case, we conjecture that wherever the Lyapunov exponent is zero, the arithmetical exponents achieve a local maximum.

Vinberg’s representations and arithmetic invariant theory

Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over ℚ as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves. In this paper we construct for each simply laced Dynkin diagram a coregular representation (G,V) and a family of algebraic curves over the geometric quotient V∕∕G. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of G. In the case of type A2, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over ℚ.

Rational orbits around charged black holes

We show that all eccentric timelike orbits in Reissner-Nordstr\"om spacetime can be classified using a taxonomy that draws upon an isomorphism between periodic orbits and the set of rational numbers. By virtue of the fact that the rationals are dense, the taxonomy can be used to approximate aperiodic orbits with periodic orbits. This may help reduce computational overhead for calculations in gravitational wave astronomy. Our dynamical systems approach enables us to study orbits for both charged and uncharged particles in spite of the fact that charged particle orbits around a charged black hole do not admit a simple one-dimensional effective potential description. Finally, we show that comparing periodic orbits in the Reissner-Nordstr\"om and Schwarzschild geometries enables us to distinguish charged and uncharged spacetimes by looking only at the orbital dynamics.

AN ARITHMETICAL INVARIANT OF ORBITS OF AFFINE ACTIONS AND ITS APPLICATION TO SIMILARITY CLASSES OF QUADRATIC SPACES

Given an action of an affine algebraic group on an affine variety and a relatively invariant regular function, all defined over the ring of integers of a number field and having suitable additional properties, an invariant of the rational orbits of the action is defined. This invariant, the reduced replete Steinitz class, takes its values in the reduced replete class group of the number field. The general framework is then applied to obtain an invariant of similarity classes of non-degenerate quadratic spaces of even rank. The invariant is related to more familiar invariants. It is shown that if the similarity classes are weighted by the volume of an associated automorphism group then their reduced replete Steinitz classes are asymptotically uniformly distributed with respect to a natural parameter.

On parameterizations of rational orbits of some forms of prehomogeneous vector spaces

In this paper, we consider the structure of rational orbit of some prehomogeneous vector spaces motivated by Wright and Yukie’s work. After we give a refinement of Wright and Yukie’s construction, we apply it to the inner forms of the D5 and E7 types. Parameterizations of reducible algebras for the cubic cases are included.

PERIODIC ORBITS OF QUADRATIC POLYNOMIALS

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

Admissible nilpotent orbits of real and 𝑝-adic split exceptional groups

We determine the admissible nilpotent coadjoint orbits of real and p p -adic split exceptional groups of types G 2 G_2 , F 4 F_4 , E 6 E_6 and E 7 E_7 . We find that all Lusztig-Spaltenstein special orbits are admissible. Moreover, there exist non-special admissible orbits, corresponding to “completely odd” orbits in Lusztig’s special pieces. In addition, we determine the number of, and representatives for, the non-even nilpotent p p -adic rational orbits of G 2 G_2 , F 4 F_4 and E 6 E_6 .

On reflection length in reflection groups

Let W be the Weyl group of a connected reductive group over a finite field. It is a consequence of the Borel-Tits rational conjugacy theorem for maximal split tori that for certain reflection subgroups W 1 of W (including all parabolic subgroups), the elements of minimal reflection length in any coset wW 1 are all conjugate, provided w normalises W 1. We prove a sharper and more general result of this nature for any finite Coxeter group. Applications include a fusion result for cosets of reflection subgroups and the counting of rational orbits of a given type in reductive Lie algebras over finite fields.

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/Gℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/Gℤ; that ifH is epimorphic inG then the action ofH onG/Gℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/Gℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.

Stability of the Barto-Sutton Network under Rational Control Orbits

The class of networks based on the Barto-Sutton architecture are known to be capable of solving complex, multi-dimensional control problems. In these problems, the objective of the task is the localization of a system within a contiguous region of its state space. In this work, asymptotic stability criteria are derived for the Adaptive Critical Element (ACE) of the network. Here, the weights of the network are viewed as the state of a linear time-variant state- space learning machine. For system trajectories which can be represented by simple rational polynomials, discrete-time techniques are used to analyze the stability of the learning machine. The advantages of this approach are that it both provides bounds for the learning parameters and characterizes the resultant learning behavior.

Rational Orbits on Three-Symmetric Products of Abelian Varieties

Let $A$ be an $n$-dimensional Abelian variety, $n \geq 2$; let ${\text {CH}_0}(A)$ be the group of zero-cycles of $A$, modulo rational equivalence; by regarding an effective, degree $k$, zero-cycle, as a point on ${S^k}(A)$ (the $k$-symmetric product of $A$), and by considering the associated rational equivalence class, we get a map $\gamma :{S^k}(A) \to {\text {CH}_0}(A)$, whose fibres are called $\gamma$-orbits. For any $n \geq 2$, in this paper we determine the maximal dimension of the $\gamma$-orbits when $k = 2$ or $3$ (it is, respectively, $1$ and $2$), and the maximal dimension of families of $\gamma$-orbits; moreover, for generic $A$, we get some refinements and in particular we show that if $\dim (A) \geq 4$, ${S^3}(A)$ does not contain any $\gamma$-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: ${A_t} = {E_t} \times B$ (${E_t}$ is an elliptic curve with varying moduli) and we have constructed suitable projections between ${S^k}({A_t})$ and ${S^k}(B)$ which preserve the dimensions of the families of $\gamma$-orbits; then we have done induction on $n$. For $n = 2$ the proof is based upon the papers of Mumford and Roitman on this topic.

Rational orbits on three-symmetric products of abelian varieties

Let A A be an n n -dimensional Abelian variety, n ≥ 2 n \geq 2 ; let CH 0 ( A ) {\text {CH}_0}(A) be the group of zero-cycles of A A , modulo rational equivalence; by regarding an effective, degree k k , zero-cycle, as a point on S k ( A ) {S^k}(A) (the k k -symmetric product of A A ), and by considering the associated rational equivalence class, we get a map γ : S k ( A ) → CH 0 ( A ) \gamma :{S^k}(A) \to {\text {CH}_0}(A) , whose fibres are called γ \gamma -orbits. For any n ≥ 2 n \geq 2 , in this paper we determine the maximal dimension of the γ \gamma -orbits when k = 2 k = 2 or 3 3 (it is, respectively, 1 1 and 2 2 ), and the maximal dimension of families of γ \gamma -orbits; moreover, for generic A A , we get some refinements and in particular we show that if dim ⁡ ( A ) ≥ 4 \dim (A) \geq 4 , S 3 ( A ) {S^3}(A) does not contain any γ \gamma -orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: A t = E t × B {A_t} = {E_t} \times B ( E t {E_t} is an elliptic curve with varying moduli) and we have constructed suitable projections between S k ( A t ) {S^k}({A_t}) and S k ( B ) {S^k}(B) which preserve the dimensions of the families of γ \gamma -orbits; then we have done induction on n n . For n = 2 n = 2 the proof is based upon the papers of Mumford and Roitman on this topic.