c 2017 Society for Industrial and Applied Mathematics

SIAM J. APPL. ALGEBRA GEOMETRY
Vol. 1, pp. 352–373

The Arithmetic Geometry of Resonant Rossby Wave Triads∗
Gene S. Kopp†

Abstract. Linear wave solutions to the Charney–Hasegawa–Mima equation with periodic boundary conditions
have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, theset of resonant triads may be described as the set of integer solutions to a particular homogeneous
Diophantine equation, or as the set of rational points on a projective surface X. The set of allresonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our
work independently finds all resonant triads via a rational parametrization of X. We provide afiberwise description of X as a rational singular elliptic surface, yielding many new results about theset of wavevectors belonging to resonant triads. In particular, we show there is an infinite numberof resonant triads (with relatively prime wavevectors) containing a wavevector (a, b) with a/b = r,
where r is any given rational, and we provide a method to find these triads. This is applied to find
all resonant Rossby wave packets that are stationary in the east-west direction.
Key words. wave turbulence theory, β-plane, Rossby wave, drift wave, Charney–Hasegawa–Mima equation,conservation of potential vorticity, resonance, arithmetic geometry, Diophantine equation, elliptic
curve, rational elliptic surface, Mordell–Weil group, Chabauty–Coleman method
AMS subject classifications. Primary, 11D41, 14G05, 76B65, 86A10; Secondary, 11D45, 11G05, 11G35, 14G25,
14M20
DOI. 10.1137/16M1077593

1. Introduction. This paper determines the set of resonant Rossby/drift wavevector triads
as the set of values of explicit rational functions of three parameters. It uses methods fromalgebraic geometry and number theory. We identify the primitive resonant triads as rational
points on a rational elliptic surface, and we prove results about the group structure on thefibers (which are elliptic curves).
1.1. Rossby waves. Atmospheric Rossby waves are large-scale meanders in high-altitude
winds caused by a planet’s rotation. They are a major influence on the weather and havebeen widely studied. Recently, Petoukhov et al. [24] implicated quasi-resonance of Rossby
waves in several extreme weather events. Coumou, Lehmann, and Beckmann [9] connected
the amplification of Rossby waves with global warming.
Mathematically, Rossby waves are modeled as linear wave solutions to a nonlinear differential equation—the Charney–Hasegawa–Mima equation (CHME) [3]—expressing conservationof potential vorticity (see [23], especially section 3.16). Rossby waves exist in the β-planemodel, in which the Coriolis force varies as a linear function of the latitude (see [23, sec∗
Received by the editors June 1, 2016; accepted for publication (in revised form) May 5, 2017; publishedelectronically July 20, 2017.
http://www.siam.org/journals/siaga/1/M107759.html
Funding: This research was supported by NSF grant DMS-1401224 and NSF RTG grant DMS-1045119.
†
Department of Mathematics, University of Michigan, Ann Arbor, MI (gkopp@umich.edu).

352

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

353

tion 3.17 and Chapter 6]). The β-plane model was introduced by Rossby et al. [25], and theCHME was formally derived in the meteorological context by Charney [5, 6]. Many years later,
Hasegawa and Mima rediscovered the CHME in a plasma physics context [12, 13]: Rossby
waves in the β-plane model are mathematically equivalent to drift waves in a plasma in atokamak.
The shape of a Rossby/drift wave is described by its zonal and meridional wavenumbers,
represented as an integer wavevector (k, `) ∈ Z2 . The zonal (east/west) wavenumber k is thenumber of crests per period (around a latitudinal circle), whereas the meridional (north/south)
wavenumber ` quantifies its meridional propagation.
Triples of Rossby waves may display resonance, in which case they are called a resonant
triad. Under certain physical conditions (large β), exact resonances dominate the behavior ofa system obeying the CHME, according to a theorem of Yamada and Yoneda [30].
Enumerating resonant triads up to some large wavenumber bound is a question of practicalimportance. As Bustamante and Hayat [3] stress in their introduction, information about highfrequency resonances is needed to describe small-scale turbulence in numerical simulations.
Moreover, mesoscale waves observed in Jupiter’s atmosphere have wavelengths of about 20 km,
and would have wavenumbers of approximately 20000 if “extended to cover a significant partof Jupiter’s surface” [3].1
The set of resonant triads may be described as the set of integer solutions to a Diophantine
equation, as explained in the next section. In this paper, we give two descriptions of thesolutions to this equation: a rational parametrization and an elliptic fibration.
1.2. The CHME and Rossby waves. The CHME is2
(1.1)

∂
∂ψ
(∆ψ − F ψ) + β+ [ψ, ∆ψ] = 0,∂t∂xwhere ψ = ψ(x, y, t) is a function on R2 × R. This first two terms are linear, and their sum iscalled the linear part. The last term is nonlinear. Standard notation is used for the Euclidean
Laplacian and the Jacobian:

 2
∂ 2
∂
∆ :=
+
,
∂x
∂y
∂f ∂g ∂f ∂g[f, g] :=
−
,
∂x ∂y
∂y ∂x


(1.2)
(1.3)

and β > 0 and F ≥ 0 are constants.3 The following are the physical meanings of the parameters: (x, y) are spatial coordinates, t is a temporal coordinate, and ψ is the streamfunction of
an incompressible flow.
1

See the NASA photo at http://photojournal.jpl.nasa.gov/catalog/PIA00724. Jupiter’s mean radius r isabout 70000 km, and the waves were observed propagating east-west at 15◦ south. So, the zonal wavenumber
would be 2πr cos (15◦ ) /(20 km) ≈ 20000 waves per period (or about 3000 waves per radian).
2∂
Sometimes an additional term U ∂x(∆ψ) for a mean zonal flow is included; including it would have noeffect on the Diophantine equation we’ll obtain for resonant wavevectors.
3
For atmospheric Rossby waves, β = 2$ rcos θ , where $ is the planet’s angular rotation, θ is the latitude,
and r is the planet’s mean radius. The constant F = R12 , where R is the Rossby deformation radius.

354

GENE S. KOPPFor the derivation of the CHME in both the meteorology and plasma physics contexts,
see [11].
We will only consider the limiting case of (1.1) to infinite Rossby deformation radius.
Equivalently, we set F = 0. So, the equation we will consider is
(1.4)

∂
∂ψ
(∆ψ) + β+ [ψ, ∆ψ] = 0.
∂t∂xEquation (1.4) has a rich set of solutions, and no general solution is known. However, one
simple family of solutions, called Rossby waves, are given by exponential functions:4
(1.5)
(1.6)

ψ = ψk,` = exp (i(kx + `y − ωt)) ,βk
ω = ωk,` = − 2.
k + `2

Indeed, Rossby waves satisfy both the linear and nonlinear parts of (1.4), separately. The
vector (k, `) ∈ R2 is called the wavevector, k is the zonal wavenumber, and ` is the meridionalwavenumber.
1.3. Resonant triads. Generally, linear combinations of Rossby waves no longer satisfy
the nonlinear part of the CHME. However, under certain conditions, triads {(k1 , `1 ), (k2 , `2 ),(k3 , `3 )} of Rossby waves “resonate” so that the real part of a certain linear combination(1.7)< (A1 (τ )ψk1 ,`1 + A2 (τ )ψk2 ,`2 + A3 (τ )ψk3 ,`3 ) ,where τ is a slow time scale, satisfies the CHME. See [17] and especially the supplement [18]for details, and [23, 22, 10] for additional background.
Resonant triads of Rossby waves satisfy the conditions(1.8)

k1 + k2 = k3 ,(1.9)

`1 + `2 = `3 ,(1.10)

ω1 + ω2 = ω3 .

We have (k2 , `2 ) = (k3 − k1 , `3 − `1 ), so the condition for {(k1 , `1 ), (k2 , `2 ), (k3 , `3 )} to be aresonant triad comes down to the condition on the angular frequencies:(1.11)

ω(k1 , `1 ) + ω(k3 − k1 , `3 − `1 ) = ω(k3 , `3 ),which we rewrite as(1.12)
4k1
k3
k1 − k3
− 2
−
= 0.
2
2
2
k1 + `1 k3 + `3 (k1 − k3 )2 + (`1 − `3 )2

The real part of ψk,` is also a solution and, in some applications, may instead be called a Rossby wave.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

355

1.4. CHME on a torus. We impose the additional requirement that the solutions to the
CHME be periodic in both spatial coordinates:(1.13)

ψ(x + 2π, y, t) = ψ(x, y, t),(1.14)

ψ(x, y + 2π, t) = ψ(x, y, t).

Equivalently, rather than considering the CHME on R2 × R, we can specify that the domain
is (R/2πZ)2 × R. Periodicity implies that the wavenumbers are integers.
Several authors framed the problem of enumerating resonant triads as a number-theoretic
question [3, 15]. For convenience, change notation and set (k1 , `1 , k3 , `3 ) = (a, b, x, y).
Question 1. Find the integer solutions (a, b, x, y) to the Diophantine equation(1.15)aa2 + b2−xx2 + y 2

−

a−x
= 0.
(a − x)2 + (b − y)2

Bustamante and Hayat [3] classify integer solutions to (1.15) by mapping them bijectively
to representations of zero by a certain quadratic form, yielding an algorithm for enumerating
all resonant triads (see further discussion in subsection 1.8). Kishimoto and Yoneda [15]show that there are no solutions where b = 0. Both sets of authors also draw attention toa special family of resonant triads (called “pure cube triads” by [3] because x/y is a perfectcube rational):(1.16)(a, b) = (s4 , st3 ),(1.17)(x − a, y − b) = (t4 − s4 , −s3 t − st3 ),(1.18)(x, y) = (t4 , −s3 t),

where s, t ∈ Z.
Other objects of interest to [15] and [30] are the wavevector set Λ and primitive wavevector
set Λ0 , as well as their distribution within the integer lattice.
(1.19)

Λ = {(a, b) ∈ Z2 : ∃(x, y) ∈ Z2 , (a, b, x, y) ∈ X, ax(a − x) 6= 0},(1.20)

Λ0 = {(a, b) ∈ Z2 : ∃(x, y) ∈ Z2 , (a, b, x, y) ∈ X, ax(a − x) 6= 0, gcd(a, b, x, y) = 1}.

The condition ax(a − x) 6= 0 imposed in (1.19) and (1.20) rules out single wave solutions andzonal resonances (see also the discussion at the end of subsection 3.1).
1.5. Results. The left-hand side of (1.15) is a homogeneous rational function in (a, b, x, y),so any rational solution (a0 , b0 , x0 , y0 ) gives rise to a primitive integer solution (λa0 , λb0 , λx0 ,λy0 ) by clearing denominators and dividing out common factors. A primitive integer solution
is one where gcd(a, b, x, y) = 1, and any integer solution is an integer multiple of a primitive
integer solution. The problem of enumerating primitive integer solutions to (1.15) is the sameas the problem of enumerating rational solutions up to a common factor, that is, rational
solutions in projective space.
We may clear denominators in (1.15) and do some algebra to obtain the equation(1.21)x(a2 + b2 )(a2 + b2 − 2ax − 2by) = a(x2 + y 2 )(x2 + y 2 − 2ax − 2by).

356

GENE S. KOPP

We consider (1.15) and (1.21) as equations in projective space P3 with homogeneous coordinates [a : b : x : y]. Let X be the locus of solutions to (1.21) in P3 ; X is a degree 5 surface.
Let X ◦ ⊂ X be the locus of solutions to (1.15).
Finding points on X is essentially the same problem as finding points on X ◦ ; X has a few
extra points that are easy to describe (see subsection 2.1).
For fixed (a, b) ∈ C2 \ (0, 0), let C(a, b) be the locus of (1.21) in affine space A2 . Over any
field containing a and b, C(a, b) ∼= C(λa, λb) by (x, y) 7→ (λx, λy), so we think of C(a, b) as
1
depending only on [a : b] ∈ P . However, we think of C(a, b) as an affine curve because wewill want to consider, for (a, b) ∈ Z2 , the integer points C(a, b)(Z) on C(a, b), and a scaling
factor may enlarge the set of integer points. We denote by C(a, b) the projective closure ofC(a, b) or, equivalently, the fiber of the map X → P1 taking [x : y : a : b] 7→ [a : b].
We show that X is a rational surface, that is, a surface with a rational parametrization(meaning, a birational equivalence with P2 defined over Q). We give the following parametrization, which may be used to directly enumerate integer points on X.
Theorem 2. The surface X is a rational surface. A rational parametrization P2 → X,
where [s : t : u] 7→ [a : b : x : y] are homogeneous coordinates, is given by
  

a
s3 t(s − 2u)..
 ..   b   s(−s2 u(s − 2u) + (t2 + u2 )(t2 − 2su + u2 )) 
 ..  ..
(1.22)
.
 =
2
2
2
2

 x  t(t + u )(t − 2su + u ) ..  
..
2
2
2
2
2
(t + u )(−s (s − 2u) + u(t − 2su + u ))y
Its rational inverse X → P2 , [a : b : x : y] 7→ [s : t : u] is given by   2sa + b2..
 ..   t  =  bx − ay  .
(1.23)
 ..  ..
uax + byThe surface X may be described more precisely as a rational singular elliptic surface,that is, a surface with a map X → P1 whose generic fiber has geometric genus 1, along with
an identity section P1 → X. The rational points (x, y) on a fiber C(a, b) are, up to scaling,
the wavenumbers that exhibit resonance with (λa, λb) for some scaling factor λ. The integer
points on C(a, b) ⊂ C(a, b) are the wavenumbers that exhibit resonance with (a, b).
The standard definition of an elliptic surface includes the conditions that the surface issmooth and the generic fiber is smooth. We call X a singular elliptic surface because these
conditions do not hold: X is nonsmooth and C(a, b) is always nonsmooth.
Theorem 3. X is a rational singular elliptic surface. The map X → P1 given in homogeneous coordinates by [x : y : a : b] 7→ [a : b] (and having fibers C(a, b)), along with the choiceof identity section [a : b] 7→ [0 : 0 : a : b], is an elliptic fibration. The singular fibers over Q
occur at [a : b] = [0 : 1], [±i : 1], [±2i : 1]. A Weierstrass form of (a smooth model for) C(a, b)
is(1.24)

W 2 = Z 3 + (a2 − 2b2 )Z 2 + (a2 + b2 )2 Z.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

357

The rational points on an elliptic curve have the structure of a finitely generated abeliangroup, the Mordell–Weil group. We prove a result about the structure of this group for C(a, b).
Theorem 4. Suppose ab 6= 0. The Mordell–Weil group of C(a, b) is of the form Z/2Z × Zr ,where the rank r ≥ 1 depends on (a, b). The point P = (a, b) has order 2 and generates thetorsion subgroup, and the point Q = (0, 2b) has infinite order.
A “typical” elliptic curve over Q(t) has trivial Mordell–Weil group, that is, rank 0 and no
torsion. Thus, Theorem 4 says that X is a special member of the class of (possibly singular)
elliptic surfaces.
For definitions and background information about elliptic curves and elliptic surfaces, seeSilverman’s books [27, 28].
1.6. Discussion. This paper describes the surface X in two different ways. The firstdescription is a parametrization of X as a rational surface. This parametrization gives a procedure for enumerating points on X (i.e., primitive resonant Rossby triads). The algorithmic
performance of this enumeration procedure is discussed in the next section.
The second description is a fiberwise description: It describes all the triads with a/b fixed.
For example, the resonant wavevectors with a/b = 1—that is, those whose wave packets
propagate due northward5 —are the rational points on C(1, 1), a rank 1 elliptic curve. This
special case is discussed further in section 4.
Our parametrization of X generalizes the family of points (1.16) by introducing a third
variable, u. Like Bustamante and Hayat’s classification (which involves a different rationalchange of variables), our parametrization provides a conjecturally fast procedure for enumerating points on X. Computational issues are discussed in subsection 1.8.
Our fiberwise description of X is new and yields new information about the primitive
wavevector set, shown in Figure 2. Specifically, the nonzero rank of C(a, b) implies that there
are infinitely many points on any line through the origin with rational slope. In other words,
for any nonzero rational number r, there are infinitely many primitive resonant wavevectors(x, y) ∈ Λ0 such that xy = r. Equivalently, there are infinitely many primitive resonant
wavevectors with any southward phase direction of rational slope.
1.7. Conjectures and further work. Kishimoto and Yoneda [15] remark that the wavenumber set appears highly anisotropic. Specifically, this set becomes sparser faster as a → ∞than as b → ∞. Based on numerical observations, we suggest some stronger conjectures about
the anisotropy.
Numerical evidence suggests that the horizontal lines contain only finitely many points inthe wavenumber set.
√
Conjecture 5. If (a, b) is in the wavenumber set Λ and a, b > 0, then a < 3b4 unless(a, b) = (800, 4). In particular, there are finitely many points on every horizontal line in Figures 1 and 2. In other words, there are only finitely many Rossby waves with fixed meridionalwavenumber that exhibit nonzonal resonances.
 β a2 −b2

(
)
2βab
The group velocity of ψa,b is Cg := ∂ω=2 ,2 , so when a/b = 1, the zonal component is
∂a
2
2
2
2
(a +b ) (a +b )zero and the meridional component is positive.
5

358

GENE S. KOPP
b
1000

500

-1000

500

-500

a
1000

-500

-1000

Figure 1. The wavevector set Λ on the domain |a|, |b| ≤ 1000.

On the other hand, it’s easy to produce infinitely many points on some vertical lines.
For example, if a = s4 , then (a, st3 ) ∈ Λ for any t ∈ Z. We conjecture, based on numericalcomputations, that there are actually infinitely many points on any vertical line (except theb-axis) in Figures 1 and 2.
Conjecture 6. For fixed nonzero a ∈ Z, {b ∈ Z : (a, b) ∈ Λ0 } is infinite.
Define the growth functions(1.25) F1 (N ) := Λ0 ∩ [−N, N ]2 ,(1.26) F2 (N ) := Λ0 ∩ (R × [−N, N ]) ,(1.27) F3 (N ) := {(a, b, x, y) ∈ X : gcd(a, b, x, y) = 1, (a, b, a − x, b − x, x, y) ∈ [−N, N ]6 } .
Numerical evidence suggests that F1 (N ), F2 (N ), and F3 (N ) all grow linearly in N , to first

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

359

b
1000

500

-1000

500

-500

a
1000

-500

-1000

Figure 2. The primitive wavevector set Λ0 on the domain |a|, |b| ≤ 1000.

order. Thus, the corresponding quantities without the primitivity condition should grow likeΘ(N log N ). We hope to investigate the growth of these quantities in a future paper.
Finally, we make a remark about the distribution of ranks of these elliptic curves. For(a, b) ∈ [1, 100]2 , we found (using Magma) that 39% of the C(a, b) had rank 1, 46% had
rank 2, 13% had rank 3, and 1% had rank 4. This looks similar to the numerical data that
number theorists have obtained for other rank 1 families. However, a natural generalizationof the widely believed Minimalist Conjecture predicts that, in the limit as N → ∞, 50% of
the C(a, b) in [1, N ]2 should have rank 1, and 50% rank 2. See [1] for background about thisproblem and extensive numerical computations.
1.8. Computational issues. Bustamante and Hayat’s classification [3] yields an algorithm
for enumerating the set of nontrivial resonant triads. They identify this set with the zero-set
of a diagonal quadratic form in four variables and enumerate that zero-set using classical

360

GENE S. KOPP

theorems on representations of integers as n = x2 + dy 2 for d = 1, 3. Bustamante and Hayat
pose the challenge of generating all or most primitive triads within a box of size N :(1.28)

BN := {(a, b, x, y) ∈ X : gcd(a, b, x, y) = 1, (a, b, a − x, b − x, x, y) ∈ [−N, N ]6 }.

For N = 5000, Bustamante and Hayat’s method produced 870 × 12 = 435 × 24 triads in 20
minutes in Mathematica on an “8-core desktop” [3]. Bustamante and Hayat compare thisto a naı̈ve search, which would take about 15 years to complete, although Kartashov andKartashova [14] remark that a smart search (confined to an open neighborhood of X) would
be faster.6
For our method, we use the parametrization (1.22) to enumerate points on X, with a slight
modification. We changed coordinates on P2 to make it easier to mod out by symmetries: We
set w = s − 2u and use the coordinates [s : t : w]. Then all nontrivial resonant triads are
generated by s > 0, t > 0, w > 0, t2 > 41 (3s − w)(s + w), and by the action of the group oforder 24 generated by the symmetries(a, b, x, y) 7→ (−a, −b, −x, −y),(a, b, x, y) 7→ (x, y, a, b),(a, b, x, y) 7→ (a, b, a − x, b − y),(a, b, x, y) 7→ (a, −b, x, −y).
Searching 0 < s, t, w ≤ 200 and applying symmetries took about 5 minutes of CPU timeand found 443 × 24 triads in B5000 . Searching 0 < s, t, w ≤ 500 and applying symmetries
took about 80 minutes and found 463 × 24 triads in B5000 . Computations were performed
in Mathematica on a MacBook Pro laptop (2.9 GHz dual-core i7). The Mathematica codefor these computations is included as an electronic supplement to the paper (M107759 01.zip[local/web 220KB]).
We have not established any upper bound on the search space necessary to output all
nontrivial resonant triads up to a given wavenumber bound. We hope to establish such abound in a future paper.
2. The surface of Rossby triads. In this section, we describe the exceptional points on
X, and we prove Theorem 2 giving a birational equivalence between X and P2 .
2.1. Exceptional sets. Our object is to determine the rational points of X ◦ —that is, theprimitive Rossby wave triads. It’s convenient to study the Zariski closure X, which will havea few extra points. Before we do so, it’s important to quantify the difference between these
two sets. We also need to understand the singular points of X, because the birational map
from X → P2 that we will define in subsection 2.2 will be undefined at these points.
Proposition 7. X \ X ◦ is a union of 3 real and 12 nonreal lines; specifically,
X \ X ◦ ={a = b = 0} ∪ {x = y = 0} ∪ {a − x = b − y = 0} ∪ {a2 + b2 = x2 + y 2 = 0}
(2.1)
6

∪ {a2 + b2 = (a − x)2 + (b − y)2 = 0} ∪ {x2 + y 2 = (a − x)2 + (b − y)2 = 0}.

Kartashov and Kartashova describe, in section IV of [14], a search algorithm enumerating BN in O(N 3 )time, compared to O(N 4 ) for a naı̈ve search.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

361

Proof. On X \ X ◦ , we have (1.21), and one of the denominators in (1.15) vanishes.
If a2 + b2 = 0, then a(x2 + y 2 )(x2 + y 2 − 2ax − 2by) = 0. There are three subcases.
(i) If a = 0, then b2 = 0, so a = b = 0.
(ii) If x2 + y 2 = 0, then a2 + b2 = x2 + y 2 = 0; this is a union of the four nonreal lines{a+bi = x+yi = 0}∪{a+bi = x−yi = 0}∪{a−bi = x+yi = 0}∪{a−bi = x−yi = 0}.
(iii) If x2 + y 2 − 2ax − 2by = 0, then a2 + b2 = (a − x)2 + (b − y)2 = 0; this is also a unionof four nonreal lines.
The two remaining cases, x2 + y 2 = 0 and (a − x)2 + (b − y)2 = 0, are similar.
The real points of X \ X ◦ correspond to “resonant triads” that are actually just single
Rossby waves. If, for example, x − a = y − b = 0, then the triad is {(a, b), (0, 0), (a, b)}, andthe conditions on the Ai (t) (see equation (5) in [18]) reduce to A01 (t) = A03 (t) = 0. Thus, (1.7)
becomes ψ = < (Cψa,b ), the real part of a single Rossby wave.
Proposition 8. The singular locus X sing of X is the algebraic set where
a2 + b2 = 0

(2.2)

and

x2 + y 2 = 0

anday − bx = 0,

that is, the two complex projective lines(2.3)[a : ia : x : ix]

and[a : −ia : x : −ix].

Proof. The surface X = {[a : b : x : y] ∈ P3 : f (a, b, x, y) = 0}, where(2.4)

f (a, b, x, y) = x(a2 + b2 )(a2 + b2 − 2ax − 2by) − a(x2 + y 2 )(x2 + y 2 − 2ax − 2by).

The singular locus is
(2.5)

X

sing


=


∂f
∂f
∂f
∂f
[a : b : x : y] ∈ P : f =
=
=
=
=0 .
∂a
∂b
∂x
∂y
3

A straightforward algebraic computation shows that this is the set described in the proposition.
Note that X sing is contained in X \ X ◦ ; that is, X ◦ is everywhere smooth.
2.2. Rational parametrization. We will now prove Theorem 2, which we restate here forconvenience.
Theorem 9. The surface X is birational to P2 over Q. A rational parametrization P2 → X,
in homogeneous coordinates [s : t : u] 7→ [a : b : x : y], is given by 
s3 t(s − 2u)a..
 ..   b   s(−s2 u(s − 2u) + (t2 + u2 )(t2 − 2su + u2 )) 
 ..  
..
.
 =
2
2
2
2

 x  
t(t
+
u
)(t
−
2su
+
u
)

 ..  
..
2
2
2
2
2
(t + u )(−s (s − 2u) + u(t − 2su + u ))
y

(2.6)

362

GENE S. KOPP

Its rational inverse X → P2 , [a : b : x : y] 7→ [s : t : u] is given by   2sa + b2..
 ..   t  =  bx − ay  .
(2.7)
 ..  ..
uax + by
Proof. Define the rational maps φ : P3 → P3 and ψ : P3 → P3 by(2.8)

φ([s : t : u : v]) = [st2 : s(sv − ut) : stv : −t3 − tu2 + suv],(2.9)

ψ([a : b : x : y]) = [a(a2 + b2 ) : a(bx − ay) : a(ax + by) : x(bx − ay)].

The map φ is defined away from the four lines(2.10){s = t = 0} ∪ {s = 0, t = iu} ∪ {s = 0, t = −iu} ∪ {t = v = 0}.

The map ψ is defined away from the four lines(2.11){a = b = 0} ∪ {a = x = 0} ∪ {a = ib, x = iy} ∪ {a = −ib, x = −iy}.

On a Zariski dense open set, φ ◦ ψ = id and ψ ◦ φ = id; that is, φ and ψ are rational
inverses. Moreover, if [a : b : x : y] = φ([s : t : u : v]) and [a : b : x : y] ∈ X, then (1.21)
becomes (after some algebra)(2.12) st(t4 + t2 u2 − 2stuv + s2 v 2 )2 (−t5 + 2st3 u − 2t3 u2 + 2stu3 − tu4 + s4 v − 2s3 uv) = 0.
But st(t4 + t2 u2 − 2stuv + s2 v 2 ) = 0 is the locus of indeterminacy for ψ ◦ φ, and so we have−t5 + 2st3 u − 2t3 u2 + 2stu3 − tu4 + s4 v − 2s3 uv = 0. Solving for v and plugging the result
into (2.8), we obtain (2.6).
3. The elliptic fibration. In this section, we show that—for all but finitely many [a : b] ∈Q —the fiber C(a, b) of X → P1 is a genus 1 curve with two singular points, and we finda smooth model C̃(a, b) in Weierstrass form. Finally, we classify the torsion points on eachC̃(a, b).

P1

3.1. Normalization of the fiber.
Proposition 10. The curve C(az, bz) in P2 (coordinates [x : y : z]) has singular points[1 : i : 0] and [1 : −i : 0].
Proof. Let g(x, y, z) = f (az, bz, x, y), where f is the polynomial (2.4) defining X, soC(az, bz) has homogeneous equation g(x, y, z) = 0. Then


∂g
∂g
∂g
(3.1)
C(az, bz)sing = [x : y : z] ∈ P2 : g =
=
=
=0 .
∂x
∂y
∂z
Solving for x, y, z, we find the two singular points [1 : i : 0] and [1 : −i : 0].
Proposition 11. If a, b ∈ R, the real points of C(a, b) form a smooth closed loop.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

Proof. Let (x, y) ∈ C(a, b)(R), and let r = 1 +
(3.2)

363


b 2
a . By (1.21),x(a2 + b2 )(a2 + b2 − 2ax − 2by) = a(x2 + y 2 )(x2 + y 2 − 2ax − 2by).

Suppose x2 + y 2 > r2 (a2 + b2 ); then, rn (a2 + b2 ) < x2 + y 2 ≤ rn+1 (a2 + b2 ) for some n ≥ 2.
Thus,
|x|(a2 + b2 )(a2 + b2 − 2ax − 2by) > rn |a|(a2 + b2 )(a2 + b2 − 2ax − 2by),|x| > rn |a|.

(3.3)
Thus,
(3.4)

r2n a2 < x2 ≤ x2 + y 2 ≤ rn+1 (a2 + b2 ) = rn+2 a2 ,so rn−2 < 1. This is impossible because r > 1 and n ≥ 2. Thus,(3.5)

x2 + y 2 ≤ r2 (a2 + b2 ).

√That is, C(a, b)(R) is contained in the closed disc around the origin of radius r a2 + b2 .
Finally, note that C(a, b) has no real singularities, by Proposition 10.
Corollary 12. If a, b ∈ Z, then C(a, b)(Z) is a finite set.
Proof. C(a, b)(Z) = C(a, b)(R) ∩ Z2 is a compact discrete subset of R2 , and hence finite.
We normalize the C(a, b) to obtain a smooth model, and convert to Weierstrass form.
Proposition 13. Consider a, b ∈ Q such that [a : b] 6= [0 : 1], [±i : 1], [±2i : 1]. Then thenormalization of C(a, b) is an elliptic curve C̃(a, b) with affine equation(3.6)

W 2 = Z 3 + (a2 − 2b2 )Z 2 + (a2 + b2 )2 Z.

A standard Weierstrass form is given by
(3.7)

1
1
Y 2 = X 3 + (2a4 + 10a2 b2 − b4 )X − (a2 − 2b2 )(7a4 + 26a2 b2 + b4 ).
3
27

Proof. Define the rational function φ by
a(a2 + b2 )2 (a − x)(a2 + b2 − ax − by + x2 + y 2 )−(a2 + b2 )2 (a − x)(3.8) φ(x, y) =,.
(a2 + b2 )x − a(x2 + y 2 )(bx − ay)((a2 + b2 )x − a(x2 + y 2 ))It is a straightforward computation to check that (Z, W ) = φ(x, y) satisfies (1.24) for generic(x, y) ∈ C(a, b), and that φ is invertible on a Zariski open set whose complement does notcontain X. The discriminant of the right-hand side of (1.24) is computed to be(3.9)∆ = −48a2 (a2 + b2 )4 (a2 + 4b2 ).

Equation (1.24) defines a nonsingular genus 1 curve whenever ∆ 6= 0, that is, whenever[a : b] 6= [0 : 1], [±i : 1], [±2i : 1].
A standard Weierstrass form comes from the substitution (X, Y ) = (Z+ 31 (a2 −2b2 ), W ).

364

GENE S. KOPP

P

T

O

P+T
Figure 3. The real points of the curve C(a, b).

We state a theorem summarizing what we have shown. Theorem 3 will follow from Theorems 14 and 15.
Theorem 14. X is a rational singular elliptic surface. The map X → P1 given in homogeneous coordinates by [x : y : a : b] 7→ [a : b] (and having fibers C(a, b)), along with the choiceof identity section [a : b] 7→ [0 : 0 : a : b], is an elliptic fibration.
Proof. This result follows from Proposition 13.
There are four “trivial” integer points on C(a, b), giving rise to physically uninteresting
solutions to the CHME. The point O = (0, 0) is taken to the identity of C̃(a, b), while
T = (a, b) has order 2. Physically, these torsion points are single Rossby waves (considered as“triads” along with two waves of amplitude zero). The point P = (0, 2b) has infinite order,and (a, −b) = P + T . Physically, P and P + T are zonal resonances.
The involution Q 7→ Q + P is represented by rotation about the center in Figure 3.
However, the negation involution Q 7→ −Q (equivalently, (Z, W ) 7→ (Z, −W ) in (1.24)) looks
much more complicated in the (x, y)-coordinates and takes P and P + T to nontrivial points
on some higher-level C(λa, λb).
3.2. Torsion points on the fiber. We will now rule out the existence of additional torsion points on C̃(a, b)(Q). This is a rather involved computation and will occupy the next
several pages.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

365

Theorem 15. For every a, b ∈ Z with a 6= 0, the torsion of C̃(a, b)(Q) is Z/2Z, generated
by T .
Proof. By a famous theorem of Mazur [19, 20], the torsion group is one of the cyclic groups
Cm for 1 ≤ m ≤ 10 or m = 12, or C2 × C2n for 1 ≤ n ≤ 4. There can be no 7-torsion because
there is 2-torsion. So, it is sufficient to rule out additional 2-torsion, 3-torsion, 4-torsion, and5-torsion. This will be done by the method of division polynomials. The 3-torsion is by far
the most difficult to rule out, and the hard work is carried out in Lemma 18.
Let ψn denote the nth division polynomial for an elliptic curve y 2 = x3 + Ax + B in
Weierstrass form. For odd n, ψn is a polynomial of A, B, and x; for even n, it is y timesa polynomial of A, B, and x. The solutions of ψn (x, y) = 0 give the locations of the nQµ(n/d)
torsion points of the elliptic curve. The primitive division polynomial ψ̂n = d|n ψd
is apolynomial of A, B, and x (for n 6= 2), and the solutions of ψ̂n (x, y) = 0 gives the x-valuesof the primitive n-torsion points, each of which has two y-values. For more information on
division polynomials (including a recursive definition), see [26, 16], as well as Exercise 3.7
in [27].
The 2-torsion: Use (1.24) for C̃(a, b),(3.10)

y 2 = x(x2 + (a2 − 2b2 )x + (a2 + b2 )2 ).

The 2-torsion points occur when y = 0, and the discriminant of the quadratic factor on theright-hand side is ∆ = −3a2 (a2 + 4b2 ) < 0. So the other two 2-torsion points are nonreal, andthus irrational.
The 3-torsion: If we set â = a2 and b̂ = b2 , then ψ3 may be written as(3.11) ψ3 = 27x4 + 18(2â2 + 10âb̂ − b̂2 )x2 − 4(â − 2b̂)(7â2 + 26âb̂ + b̂2 ) − (2â2 + 10âb̂ − b̂2 )2 .
The curve {ψ3 (a, b, x) = 0} ⊆ P(1, 1, 2) is a fourfold ramified cover of the curve {ψ3 (â, b̂, x) =0} ⊆ P2 , and this latter curve is birational to P1 (with homogeneous coordinates [k : `]).
(3.12)(3.13)[â, b̂, x] 7→ [b̂ + x : 4â + b̂ − 3x],[k, `] 7→ [64k`3 : 2187k 4 − 162k 2 `2 − 40k`3 − `4 : (27k 2 − 18k` − `2 )2 ].

Let (x, y) be a rational 3-torsion point on C̃(a, b). From the latter birational equivalence, we
obtain(3.14)

a2 = 64λk`3 ,(3.15)

b2 = λ(2187k 4 − 162k 2 `2 − 40k`3 − `4 ),(3.16)x = λ(27k 2 − 18k` − `2 )2for relatively prime integers k, ` and some λ ∈ Q. Dividing the first two equations, we obtain


 5
 3
 2  
8bk 2
k
k
k
k
(3.17)= 2187
− 162
− 40
−
.
a`
`
`
`
`
This looks like the equation of a hyperelliptic curve, which we prove has only two rational
points in Lemma 18. By that lemma, [k : `] is either [0 : 1] or [1 : 0]. Thus, a = 64λk`3 = 0,
a contradiction. So there are no rational 3-torsion points.

366

GENE S. KOPP

The 4-torsion: The primitive 4-division polynomial factors as(3.18)

ψ̂4 =

1
ψ4
=
(3x − (4a2 + b2 ))(3x + (2a2 + 5b2 ))p(x),
ψ2
729

where
(3.19)

p(x) = 81x4 + 54(a2 − 2b2 )x3 + 54(7a4 + 26a2 b2 + b4 )x2(3.20)− 6(a2 − 2b2 )(20a4 + 82a2 b2 − b4 )x(3.21)+ (76a8 + 364a6 b2 + 366a4 b4 + 484a2 b6 + b8 ).

There are four possibly rational 4-torsion points R, those such that 2R = T , where T is thenontrivial rational 2-torsion point. Each such point must be defined over a quadratic extensionof Q(a, b); therefore, they must come from the first two factors of ψ̂4 . These points are


√
1
2
2
2
2
(3.22)(x, y) =(4a + b ), ± 3a(a + b ) and
3


p
1
2
2
2
2
2
2
(3.23)(x, y) = − (2a + 5b ), ±i(a + b ) a + 4b .
3Both have irrational y-values for all a, b ∈ Q.
The 5-torsion: Setting c = a2 − x and d = b2 + x, ψ5 may be written as(3.24)

312 ψ5 = 23328(c − 2d)5 (82c4 + 316c3 d + 510c2 d2 + 292cd3 + 97d4 )x3(3.25)− 1296(c − 2d)4 (2740c6 + 21552c5 d + 60396c4 d2 + 79676c3 d3+ 40623c2 d4 + 12894cd5 + 157d6 )x2(3.26)(3.27)+ 36(c − 2d)3 (69296c8 + 824032c7 d + 3756944c6 d2 + 8338240c5 d3 + 9252272c4 d4+ 4279360c3 d5 + 1262552c2 d6 + 40744cd7 + 803d8 )x(3.28)− (430016c12 + 3828480c11 d − 1707456c10 d2 − 125933632c9 d3 − 526043088c8 d4(3.29)− 830657952c7 d5 − 223114416c6 d6 + 677936160c5 d7 + 406003284c4 d8(3.30)+ 155427176c3 d9 + 7889280c2 d10 + 216348cd11 − 409d12 )The curve {ψ5 (c, d, x) = 0} ⊆ P2 may be checked (by computer, in Magma) to be geometricgenus 1, with singular points [c : d : x] = [0 : 0 : 1], [−1 : 4 : 1], [−5 : 2 : 1]. An elliptic model
is given by(3.31)Y 2 Z + Y Z 2 = X 3 − X 2 Z,

with [0 : 0 : 1] 7→ [0 : 1 : 0] = ∞. This elliptic curve has Mordell–Weil group Z/5Z (ascomputed by Magma). The five rational points are [0 : 1 : 0], [0 : 0 : 1], [1 : −1 : 1], [1 : 0 : 1],
and [0 : −1 : 1]. They correspond to the points [0 : 0 : 1], [−5 : 2 : 1], [−1 : 4 : 1], [−5 : 2 : 1],
and [−1 : 4 : 1] on {ψ5 (c, d, x) = 0}, respectively. (We see [−1 : 4 : 1] and [−5 : 2 : 1]
twice because we’ve resolved a cuspidal singularity at [0 : 0 : 1] and nodes at [−1 : 4 : 1]
and [−5 : 2 : 1].) If [c : d : x] = [0 : 0 : 1], then [a : b] = [±i : 1], an irrational point. If[c : d : x] = [−1 : 4 : 1], then [a : b] = [0 : 1], and a = 0. If [c : d : x] = [−5 : 2 : 1], then[a : b] = [±2i : 1], an irrational point.

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

367

Corollary 16. The only resonant Rossby wave triads corresponding to torsion points on any
C̃(a, b) are single wave solutions to the CHME.
Proof. This result is just a restatement of Theorem 15.
The 3-torsion case comes down to determining the exact set of rational points on a particular hyperelliptic curve. This requires more advanced methods than are used in the rest of
this paper. An overview of the methods available for enumerating rational points on curves isgiven by Stoll [29], whereas the method of Chabauty and Coleman in particular is detailed in[21]. There are also the original papers of Chabauty [4] (in French) and Coleman [8, 7]. We
state Coleman’s theorem for easy reference (Theorem 5.3 in [21]); see [21] and the references
therein for the definition of the Coleman integral and other notation and terms.
Theorem 17 (Coleman). Let X be a curve of genus g ≥ 2 over Q, and let J = Jac(X). Let
p be a prime of good reduction for X, let r = rk J(Q), and let r0 ≤ r be the dimension of theclosure of J(Q) in J(Qp ) as a p-adic manifold. Assume that r0 < g.
(a) Let ω be a nonzero 1-form on H 0 (XQp , Ω1 ) satisfying the condition that if Qi , Q0i ∈P
P R Q0X(Qp ) such that [ i (Q0i − Qi )] ∈ J(Q), then i Qii ω = 0. Such an ω necessarily
exists, and we may assume by scaling that it reduces to a nonzero 1-form in ω ∈H 0 (XFp , Ω1 ). Suppose Q ∈ X(Fp ), and let m = ordQ ω. If m < p − 2, then the numberof points in X(Q) reducing to Q is a most m + 1.
(b) If p > 2g, then #X(Q) ≤ #X(Fp ) + (2g − 2).
The Chabauty–Coleman method, augmented by the Mordell–Weil sieve, is implemented
in the computer algebra system Magma [2], and this implementation may be used to prove
Lemma 18. Magma code doing that computation is included as an electronic supplement to
the paper (M107759 01.zip [local/web 220KB]). Magma uses local information at the primes
7, 11, 53, and 67 and makes use of the Mordell–Weil sieve.
I provide my own proof using only local data at p = 7, which I hope will be valuable
for readers interested in learning about the Chabauty–Coleman method. My proof does relyon Magma for some subcomputations, most importantly to bound the rank of the Jacobian
variety; code for these computations is also included in the electronic supplement. The Magma
handbook7 describes the algorithms used, primarily n-descent as described by Stoll [29].
Lemma 18. The hyperelliptic curve C with affine equation y 2 = 2187x5 − 162x3 − 40x2 − x
has only two rational points, Q1 = (0, 0) and Q2 = ∞.
Proof. In the first part of the proof, we will bound the number of rational points in eachcongruence class modulo 7 by the method of Chabauty and Coleman. In the second part ofthe proof, we rule out some points using a descent argument.
We begin with the method of Chabauty and Coleman. The discriminant of 2187x5 −
162x3 − 40x2 − x is −220 317 , so C has good reduction away from 2 and 3. We use the prime
p = 7 throughout.
Let J = Jac(C), let r be the rank of J(Q), and let g be the genus of C. Chabauty
will apply if r < g. The genus of a hyperelliptic curve of odd degree d is (d − 1)/2, so
g = 2. A computation in Magma shows that r = 1. Specifically, RankBounds(J) returns
7

Available online at http://magma.maths.usyd.edu.au/magma/handbook/.

368

GENE S. KOPP

0 ≤ r ≤ 1, TorsionSubgroup(J) returns Z/2Z, and Points(J:Bound:=1000)
returns several
√
√
1
2
1
pointsincluding
P
:=
[Q
−
Q
]
and
R
:=
[(
(−4
+
−11),
(7
+
5
−11))
+ ( 27
(−4 −
1
2
27
81
√
√
2
−11), 81 (7 − 5 −11)) − 2Q2 ]. The point P has order 2, so R has infinite order.
C(F7 ) has four rational points: (0, 0), ∞, (6, 1), and (6, 6). A straightforward applicationof part (b) of Theorem 17 gives#C(Q) ≤ #C(F7 ) + (2g − 2) = 6.

(3.32)To show #C(Q) = 2, we have to do more work.
We would like to construct a differentialRω ∈ H 0 (JQ7 , Ω1 ) ∼= H 0 (XQ7 , Ω1 ) such that R ωR = 0. But RR is not defined over Q7 , so wecompute with 42R instead, which is. Note that 42R ω = 42 R ω.
The point 42R = [Q01 + Q02 − Q1 − Q2 ] = [Q01 − Q1 ] + [Q02 − Q2 ], where Q01 and Q02 aredefined over Q7 :(3.33)

Q01 = (5(7)2 + 1(7)3 + 1(7)4 + · · · , 4(7) + 3(7)2 + 1(7)3 + · · · ),(3.34)

Q02 = (6(7)−2 + 0(7)−1 + 3 + · · · , 5(7)−5 + 0(7)−4 + 2(7)−3 + · · · ).

Let ω be a differential satisfying theRhypotheses of part (a) of Theorem 17; ω is a Z7 -linear
xdx
combination of dx
y and y satisfying 42R ω = 0, or, equivalently, by the properties of the
Coleman integral,
Z Q0

Z Q0

1

(3.35)2

ω+Q1

ω = 0.
Q2

Because Q1 and Q01 have the same reduction modulo 7, as have Q2 and Q02 , the relevant p-adic
integrals may be computed by antidifferentiation:
Z Q0

1(3.36)
Q1dx=y

Z 4(7)+3(7)2 +1(7)3 +···(−2 − 160y 2 − 18228y 4 − · · · )dy0(3.37)

 4(7)+3(7)2 +1(7)3 +···
160 3 18228 5= −2y −y −y − ···
35
y=0(3.38)= 6(7) + 6(7)2 + 1(7)3 + · · · ,Z Q01(3.39)
Q1xdx=y

Z 4(7)+3(7)2 +1(7)3 +···


(3.40)(3.41)=(2y 2 + 240y 4 + 30704y 6 + · · · )dy0

2 330704 7y + 48y 5 +y + ···
37
3

4

 4(7)+3(7)2 +1(7)3 +···y=05

= 3(7) + 6(7) + 6(7) + · · · .

zdz
xdx
dz
Local coordinates at ∞ are given by z = x1 and w = xy3 . We have dx
y = − y and y = − y .

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

369

Thus,
Z Q0

2

(3.42)Q2dx=y

Z 2(7)+5(72 )+4(73 )+···(−2 · 3−14 w2 − 16 · 3−31 w6 − · · · )dw

0



−15

−2 · 316 −31 7· 3 w − ···w −7
3(3.43)=(3.44)= 2(7)3 + 1(7)4 + 0(7)5 + · · · ,Z Q02(3.45)Q2xdx=y

Z 2(7)+5(72 )+4(73 )+···(3.47)

w=0(−2 · 3−7 − 4 · 3−23 w4 − · · · )dw

0
(3.46)

 2(7)+5(72 )+4(73 )+···=−2 · 3−7 w −2

4 −23 5· 3 w − ···5 2(7)+5(72 )+4(73 )+···
w=0

3= (7) + 2(7) + 0(7) + · · · .

Summing,
Z Q0
2 dx
dx
+
= 6(7) + 6(7)2 + 3(7)3 + · · · and
y
y
Q2
Q1
Z Q0
Z Q0
1 xdx
2 xdx
+
= (7) + 2(7)2 + 3(7)3 + · · · .
y
y
Q1
Q2
Z Q0

1

(3.48)(3.49)By (3.35), we must have (up to an irrelevant scalar in F×7)
ω≡(3.50)

dx xdx+(mod 7) .
yy= − (z+1)dz
has order of vanishing m = 0 at (0, 0) and ∞, andThe differential ω = (1+x)dxyw
m = 1 at (6, 1) and (6, 6). By part (a) of Theorem 17, C(Q) contains at most one point in eachof the congruence classes of (0, 0) and ∞, and at most two points in each of the congruenceclasses of (6, 1) and (6, 6). Since we’ve already found Q1 = (0, 0) and Q2 = ∞, all that remains
is to rule out the classes of (6, 1) and (6, 6).
Assume x ≡ 6 (mod 7). By factoring the right-hand side of the equation of C,
y 2 = x(2187x4 − 162x2 − 40x − 1).

(3.51)

If we write x = xx21 as a fraction in simplest from, with x1 , x2 relatively prime integers, then
(3.52)

x32 y

2


= x1 x2 2187x41 − 162x21 x22 − 40x1 x32 − x42 .


For any prime p | x1 , it follows that p - x2 and p - 2187x41 − 162x21 x22 − 40x1 x32 − x42 , so vp (x1 )must be even. For any prime p | x2 such that
p 6= 3 (note that 2187 = 37 ), it follows that p - x1
and p - 2187x41 − 162x21 x22 − 40x1 x32 − x42 . Thus, either x and (2187x4 − 162x2 − 40x − 1) are

370

GENE S. KOPPboth rational squares, or they are both rational squares divided by 3. But 67 = −1, so x is
not a square. Thus, writing (2187x4 − 162x2 − 40x − 1) = v 2 /3 and x = u/3 = w2 /3,(3.53)

v 2 = 81u4 − 54u2 − 40u − 3.

Call D the smooth completion of the curve defined by this equation. Under the birationaltransformation φ : D ∼→ E given by(3.54)(3.55)9s = (9u2 − v − 1),
81
t = (729u3 − 243u − 81uv − 98),
16

this equation becomes the elliptic curve E defined by(3.56)

t2 + t = s3 + 20.

The Mordell–Weil group of E is generated by the points P1 = (0, 4), of order 3, and P2 =(−2, 3), of infinite order. E(F7 ) has order 12, but the reductions P1 and P2 have orders 3
and 2, respectively. So, the subgroup hP1 , P2 i generated by P1 and P2 is not all of E(F7 );specifically,(3.57)

hP1 , P2 i = {∞, (4, 6), (0, 4), (5, 3), (0, 2), (4, 0)}.

Converting back to coordinates on D via φ−1 , all rational points on D fall into the congruenceclasses(3.58)
φ−1 hP1 , P2 i = {∞− , ∞+ , (6, 3), (3, 4), (3, 2), (6, 0)}.
But we’ve restricted ourselves to the case where x ≡ 6 (mod 7), that is, u = 3x ≡ 4 (mod 7),
so it’s impossible to have u ≡ ∞, 3, 6 (mod 7), and there are no additional rational points
on C.
4. Special case: Resonant wavevectors with zonal group velocity zero. Rossby waves
have phase velocity vector C and group velocity vector Cg , whereω ω  
β
βk/`
(4.1)
C = (Cx , Cy ) =
,
= − 2
,−
,
k `
k + `2 k 2 + `2!

β k 2 − `2
∂ω ∂ω
2βk`
(4.2)
Cg = (Cgx , Cgy ) =
.
,
=
,
∂k ∂`
(k 2 + `2 )2 (k 2 + `2 )2
See section 3.19 of [23] for details.
Assume k` 6= 0, and recall that β > 0. The fact that the zonal component Cx of C isnegative means that crests of Rossby waves travel westward. However, the zonal component
Cgx of Cg can be either positive, zero, or negative. If Cgx = 0, wave packets stay in the samelongitude (propagating either due northward or due southward).
We ask, what resonant wavevectors have zonal group velocity zero—that is, what wave
packets stationary in the east-west direction with nontrivial resonant interactions are possible?

ARITHMETIC GEOMETRY OF RESONANT ROSSBY WAVE TRIADS

371

All wavevectors (k, `) with Cgx = 0 are exactly those with k 2 − `2 = 0, that is, k = ±`. By
symmetry, it suffices to consider (k, `) = (n, n) with n > 0.
On C(1, 1), there are only the four trivial integer points 0 = (0, 0), T = (1, 1), P = (0, 2),
and T + P = (1, −1). By a computation in Magma (included in the electronic supplement
M107759 01.zip [local/web 220KB]), we find that C(1, 1) has rank 1, and its Mordell–Weilgroup is generated by T and P . After generators are found, it is easy to enumerate the rational
points on an elliptic curve. The first few such points on C(1, 1) are shown in Table 1.
Table 1
The first few rational points on C(1, 1), computed in Magma. If ( xn1 , yn1 ) and ( xn2 , yn2 ) are the (x, y)coordinates of kP and kP + T as shown in the third column, with xi , yi , n ∈ Z, then ( xn1 , yn1 ) + ( xn2 , yn2 ) = (1, 1),
and {(x1 , y1 ), (x2 , y2 ), (n, n)} is a resonant triad.
Point(Z, W )

(x, y) exactly(x, y) numerically0∞(0, 0)(0.000, 0.000)

T(0, 0)(1, 1)(1.000, 1.000)P(1, 2)(0, 2)(0.000, 2.000)

P +T(4, −8)(1.000, −1.000)−P(1, −2)(1, −1)

16 2
,
13 13

3 11
, 13
− 13

256 88,229 229
27 141
− 229
, 229

1792 6568,
3277 3277

1485
3291
, − 32773277
5488, − 504613
504613(0.011, −0.293)−P + T(4, 8)2P

993, − 6416

2P + T

64 496
, 27
9

−2P

9 93
,
16 64

−2P + T

64
, 27
9

3P








496

3025
,−
49


165110
84056

3P + T(−0.231, 0.846)(1.118, 0.384)(−0.118, 0.616)(0.547, 2.004)(0.453, −1.004)

147742343

196,3025 166375(1.231, 0.154)

499125 652355,
504613 504613

(0.989, 1.293)−3P

3025 165110
, 343
49



102076835570402− 155870857, 155870857−3P + T196
84056
, − 166375
3025

156891625 120300455,
155870857 155870857

4P

889249 1164191023,
553536 411830784

4P + T

2214144, − 3675053536
889249838561807−4P−4P + T


1164191023

889249, − 411830784
553536


3675053536

2214144, 838561807
889249(−0.007, 0.228)



(1.007, 0.772)

11531261952 55149444912− 34589637433, 34589637433

(−0.333, 1.594)

46120899385
20559807479, − 34589637433
34589637433

(1.333, −0.594)

9603665729328

79003970279424, − 58803854910601
58803854910601(1.343, −0.163)


68407520639929(−0.344, 1.163)− 20200115368823,
58803854910601 58803854910601

The sequence8 13, 229, 3277, 504613, 155870857, 34589637433, 58803854910601, . . . of thedenominators of the (x, y) in Table 1 exhaust the natural numbers n so that (n, n) ∈ Λ0 .
The resonant wavevectors with zonal group velocity zero are those of the form (mn, mn) and(mn, −mn), where m ∈ Z and n is in this sequence. To our knowledge, we have the first8
We have not proved that this is an increasing sequence—that is, it is theoretically possible that somehigher kP or kP + T will produce an (x, y) with smaller denominators than those obtained from some previous
k. We suspect that this does not happen, and we plan to apply the theory of height functions to this questionand others about the distribution of wavevectors in future work.

372

GENE S. KOPP

efficient method for computing the resonant wavevectors of the form (n, n), and this paper isthe first place in the literature where any have been computed beyond n = 3277.
Acknowledgments. Thank you to Jeffrey C. Lagarias for referring the author to the paperof Kishimoto and Yoneda [15], and to Charles R. Doering and Jeffrey C. Lagarias for reading
drafts of this paper. Thank you to Miguel Bustamante, Charles R. Doering, Zaher Hani,
Jeffrey C. Lagarias, and Wei Ho for helpful conversations and correspondences. Thank you
to the anonymous referees for many helpful comments.
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