Periodic continued fractions and hyperelliptic curves

0.21 MB·English

by  M-P. Grosset

PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC
7 CURVES
0
0
2 M-P.GROSSETANDA.P.VESELOV

n
a
J

1
3 Abstract. We investigate when an algebraic function of the form φ(λ) =
M] c−oBeffi(λAc)i+(eλ√n)tRs(iλn),Cw,chaenrebRe(wλr)ititsenaapsoalypneormioiadlicoαf-ofrdadctdioengroefetNhe=for2mg+1 with
math.G φ(λ)=[b0;b1,b2,...,bN]α=b0+ b1+ b2+...bNλ−−1+λα−bN1α+2bλ1−+λαb−Nλ2α−+1α..2. ,
[ for some fixed sequence αi. We show that this problem has anatural answer
givenbytheclassicaltheoryofhyperellipticcurvesandtheirJacobivarieties.

1 We also consider pure periodic α-fraction expansions corresponding to the
v specialcasewhenbN =b0.
2
3
9
1 1. Introduction
0
7 Consider the following continued fraction, which we will call α-fractions:
0
λ α
h/ (1) φ=b0+ b +−λ−1α2 =[b0,b1,...,]α,
at 1 b2+...
m
: where α = (α ),α C is a given sequence, b are arbitrary complex numbers, λ
v i i ∈ i
is a formal parameter. In this paper we will consider a special case of N-periodic
i
X α-fractions, when the sequences α and b are periodic with period N :
i i
r
a
α =α , b =b
i+N i i+N i
for all i 1:
≥
(2) φ=[b ;b ,b ,...,b ] .
0 1 2 N α
In the particular case when b =b we have φ=[b ,b ,...,b ] , which will be
N 0 0 1 N−1 α
called a pure N-periodic α-fraction.
This kind of fractions naturally appear in the theory of integrable systems, in
particularinthe theoryofperiodic dressingchain[1],but to the bestofourknowl-
edgehasnotbeenstudiedsofar. Wewerepartlyinspiredbyourrecentdiscussions
with Vassilis Papageorgiou on the discrete KdV equation where such continued
fractions appear as well [2].
1
2 M-P.GROSSETANDA.P.VESELOV
Because of periodicity we can write formally (2) as
λ α
φ=b + − 1 ,
0 b + λ−α2
1 b2+...bN−1+bNλ−−bα0N+φ
which implies a quadratic relation
(3) A(λ)φ2+2B(λ)φ+C(λ)=0,
whereA,B,C arecertainpolynomialsinλwithcoefficientspolynomiallydepending
on b . Thus to any periodic α-fraction (2) corresponds an algebraic function
i
B(λ)+ R(λ)
(4) φ(λ)= − ,
A(λp)
where
(5) R(λ)=B(λ)2 A(λ)C(λ)
−
is the discriminant of (3). In that case we will say that (2) is a periodic α-fraction
expansionofthealgebraicfunction(4)fromthehyperellipticextensionC(λ, R(λ))
of the field of rational functions C(λ). We leave the question of convergenpce aside
concentrating on algebraic and geometric aspects of the problem.
We will discuss the following three main questions.
Question 1. Whichalgebraicfunctions (4)admitN-periodic α-fractionexpan-
sions ?
Question2. Howmanysuchexpansionsmayexistforagivenalgebraicfunction
(4) and how to find them ?
Question 3. Whatisthe geometryofthe setoffunctions (4)fromgivenhyper-
elliptic extension (i.e. with fixed R), which admit periodic α-fraction expansions?
The answers depend on the parity of N. In this paper we will restrict our study
to the case of odd period N = 2g+1, which is the most interesting one (cf. [1]).
We will also assume that all the parameters α are distinct.
i
Note that when N (which is also the degree of R(λ)) is even, one can consider
the usual continued fraction expansions going back to Abel and Chebyshev who
discovered their relation with the classical problem of integration in elementary
functions (see a nicely written paper by van der Poorten and Tran [3] for details).
These expansions are more natural, but can not be used in the odd degree case.
We would like to mention alsothat in the classicalnumber-theoretic versionthe
answer to Question 1 is due to Galois, who proved that a quadratic irrationality
ξ = p+q√d, p and q rational numbers, d integer, has a pure periodic continued
fractionexpansionξ =[a ,...,a ]ifandonlyifξ islargerthan1anditsconjugate
0 k
ξ¯ = p q√d lies between 1 and 0 (see e.g. [4]). Note also that the periodic
− −
continued fractions of the form [a ;a ,...,a ] also appear naturally in number
0 1 k
theory as expansions of √d (see [4]).
To explain our main results let us introduce the polynomial
N
(6) A(λ)= (λ α )
i
−
iY=1
and call a polynomial R of degree N α-admissible if
(7) R(λ)=S2(λ)+A(λ)
PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES 3
for some polynomial S(λ) of degree g or less, where as before N =2g+1. We will
call a polynomial monic if its highest coefficient is 1 and anti-monic if it is equal
to 1. Note that α-admissible polynomials R are automatically monic.
−
Theorem 1. The algebraic functions φ(λ) admitting an N-periodic α-fraction ex-
pansion have the form (3, 4) with the polynomials A, B, C satisfying the following
conditions:
(1) degB g, A(λ) and C(λ) are monic and anti-monic polynomials of de-
≤
gree g and g+1 respectively
(2) the discriminant R(λ)=B2 AC is α-admissible.
−
Conversely, for an open dense subset of such triples (A,B,C) the corresponding
function (4) has exactly two N-periodic α-fraction expansions. The corresponding
b are rational functions of both coefficients of A, B, C and parameters α and can
i i
be found by an effective matrix factorisation procedure.
In the pure periodic case the only additional requirement is
(8) C(α )=0,
N
under which the pure periodic α-fraction expansion is generically unique.
Wewillcall(A,B,C)satisfyingtheconditions(1),(2)ofTheorem1theα-triples.
Note that these conditions are invariant under any permutation of the parameters
α . In fact there is a natural birational action of the direct product G = Z S
i 2 N
onthe setof periodic continued α-fractionexpansions,where the generatorε×of Z
2
isactingsimply byswappingtwodifferentα-fractionexpansionsgivenbyTheorem
1. Our next result describes this action explicitly.
Let us introduce the following permutation π S , which reverses the order
N
∈
α ,α ,...,α , α to α , α ,...,α ,α and the involutions σ swapping α
1 2 N−1 N N N−1 2 1 k k
and α , where k =1,...,N 1.
k+1
−
We will show that σ is acting on b=(b ), i=0,...,N with b =0 as follows:
k i k
6
α α α α
(9) ˜b =b + k+1− k, ˜b =b k+1− k,
k−1 k−1 b k+1 k+1− b
k k
the rest of b remain the same. This determines the action of the symmetric group
i
S since σ generate it. To describe the action of Z it is enough to describe the
N k 2
action of the involution επ G, which turns out to be quite simple:
∈
(10) ˜b = b , j =1,...,N 1 ˜b =b b , ˜b = b .
j N−j 0 0 N N N
− − − −
Theorem2. Theformulae(9)and(10)defineabirationalactionofthegroupG=
Z S on the set of N-periodic α-fractions. Its orbits consist of all 2N! possible
2 N
×
periodic α-fraction expansions for a given α-triple (A,B,C) and any permutation
of the parameters α .
i
In the pure periodic case the symmetry group is broken down to S generated
N−1
by σ with k =1,...,N 2 given by (9).
k
−
Let us fix now the α-admissible polynomial R(λ) with distinct roots. We would
liketodescribethegeometryofthesetofelementsfromthehyperellipticextension
field C(λ, R(λ)) which have a pure periodic α-fraction expansion. For the stan-
dard mateprial from the algebraic geometry of the curves we refer to the classical
Griffiths-Harris book [5].
Consider the hyperelliptic curve Γ given by the equation
R
(11) µ2 =R(λ).
4 M-P.GROSSETANDA.P.VESELOV
ThecurveΓ consistsoftheaffinepartΓaff,correspondingtothe”finite”solutions
R R
of (11), and the ”infinity” point, which we will denote as P . Since the roots of R
∞
aredistinctitisnon-singularandhasgenusg.Considerg pointsP ,...,P ofΓaff
1 g R
and call the corresponding divisor D =P + +P non-special if
1 g
···
(12) P =τ(P )
i j
6
for any i=j, where τ is the hyperelliptic involution:
6
τ(µ,λ)=( µ,λ).
−
Non-special divisors have the property that the linear space L(D) of all meromor-
phic functions on Γ having the poles at D of order less than or equal to 1 has
R
dimension 1, which means that it consists only of constant functions. The corre-
sponding linear space L(D+P ) has dimension 2, so there exists a non-constant
∞
function f L(D+P ) with additional pole at infinity. These functions are in
∞
∈
a way ”least singular” among the ”generic” meromorphic functions on Γ in the
R
sense that any such function can not have less than g+1 poles (see [5]).
Now define the affine Jacobi variety J(Γ )aff as the set of positive non-special
R
divisors D =P + +P , P ,...,P Γaff.
1 ··· g 1 g ∈ R
Let Mα be an affine variety of α-triples of polynomials (A, B, C) with given
R
discriminant R(λ), and Pα be its subvariety given by the additional condition (8).
R
Theorem 3. There exists a bijection between the set Mα and the extended affine
R
JacobivarietyJ(Γ )aff C.Thecorrespondingalgebraic functions(4)canbechar-
R
×
acterised as meromorphic functions φ L(D+P ) on Γ with non-special pole
∞ R
∈
divisor D+P and asymptotic φ √λ at infinity.
∞
∼
In the pure periodic case under the assumption that R(α ) = 0 there exists a
N
6
natural 2 : 1 map from the set Pα to J(Γ )aff. The corresponding φ from L(D+
R R
P ) are fixed by the condition that one of two values of φ(α ) is zero.
∞ N
TheproofisbasedontheclassicaldescriptionoftheJacobivarietyduetoJacobi
himself [6] (see also Mumford [7]).
We see that the affine space CN of all functions (4) having periodic α-fraction
expansion is birationally equivalent to the double covering of the bundle of the
extended affine Jacobians of α-admissible hyperelliptic curves. This is a version of
the well-knownresult by Dubrovinand Novikov[8] who were the first to apply the
theory of the KdV equation to the problems of algebraic geometry.
2. Periodic α-fractions
Consider the N-periodic α-fraction of period N =2g+1
λ α
φ=b + − 1 =[b ;b ,...,b ,b ]
0 b + λ−α2 0 1 N−1 N α
1 b2+...bN−1+bN+bλ1−+λαb−Nλ2α−+1α..2.
PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES 5
We see that (at leastformally)φ is the fixed point ofthe fractionallinear trans-
formation
a
(13) s(φ)=b + 1
0 b + a2
1 b2+...bN−1+b∗NaN+φ
with b∗ =b b and a =λ α , k =1,..,N. The function s(φ) can be written
N N − 0 k − k
as s(φ)= PN−1φ+PN , where the quantities P , Q are determined by the standard
QN−1φ+QN k k
recurrence relations (see e.g. [9], page 14):
P =1, Q =0,
−1 −1
P =b , Q =1,
0 0 0
(14) P =b P +a P , Q =b Q +a Q
k+1 k+1 k k+1 k−1 k+1 k+1 k k+1 k−1
for k <N 1 and
−
P =b∗ P +a P , Q =b∗ Q +a Q .
N N N−1 N N−2 N N N−1 N N−2
Thus we have
P φ+P
(15) φ= N−1 N ,
Q φ+Q
N−1 N
which can be written as a quadratic equation
(16) Q φ2+(Q P )φ P =0.
N−1 N N−1 N
− −
It is easy to see fromthe recurrencerelations that P and Q are polynomials in λ
k k
of the form
P =(b +b +...+b )λk+..., P =λk+1+...,
2k 0 2 2k 2k+1
Q =λk+..., Q =(b +b +...+b )λk−1+...,
2k 2k−1 1 3 2k−1
for k g and Q =(b +b +...+b +b b )λg +..., where the dots
2g+1 1 3 2g−1 2g+1 0
≤ −
denote the lower degree terms. Since from (16)
1
(17) A(λ)=Q (λ), B(λ)= (Q (λ) P (λ)), C(λ)= P (λ),
N−1 N N−1 N
2 − −
the polynomial A is monic of degree g, C is anti-monic of degree g+1 and B has
degree g or less with the highest term βλg, where
N
1
β = b + ( 1)k+1b .
0 k
− 2 −
kX=1
Let us show now that the discriminant R=B2 AC is α-admissible. We have
−
1 1
R= (Q P )2+P Q = (P +Q )2+P Q P Q .
N N−1 N N−1 N−1 N N N−1 N−1 N
4 − 4 −
We claim that
N
P Q P Q = (λ α ).
N N−1 N−1 N i
− −
iY=1
Indeed, the determinant
P P b P +a P P
N N−1 = N N−1 N N−2 N−1 =
(cid:12)(cid:12) QN QN−1 (cid:12)(cid:12) (cid:12)(cid:12) bNQN−1+aNQN−2 QN−1 (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
6 M-P.GROSSETANDA.P.VESELOV
P P b 1
a N−1 N−2 = =( 1)Na a ...a 0 .
− N(cid:12) QN−1 QN−2 (cid:12) ··· − N N−1 1(cid:12) 1 0 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Since N is odd,(cid:12) (cid:12) (cid:12) (cid:12)
N
P P
N N−1 =a a ...a = (λ α )=A.
(cid:12)(cid:12) QN QN−1 (cid:12)(cid:12) N N−1 1 Yi=1 − i
(cid:12) (cid:12)
(cid:12) (cid:12)
Now by taking S(λ) = PN−1+QN, which is a polynomial of degree g or less, we
2
seethatR(λ)=S2+A,soRisα-admissible. ThisprovesthefirstpartofTheorem
1 in the periodic case.
To prove the second part let us introduce the following matrix
1 b 0 λ α 0 λ α
(18) M(λ)= 0 − 1 ... − N ,
(cid:20) 0 1 (cid:21)(cid:20) 1 b1 (cid:21) (cid:20) 1 b∗N (cid:21)
with b∗ =b b . One can check that it can be rewritten also as
N N − 0
b λ α b λ α 1 b∗
(19) M = 0 − 1 ... N−1 − N N .
(cid:20) 1 0 (cid:21) (cid:20) 1 0 (cid:21)(cid:20) 0 1 (cid:21)
The following Lemma explains its importance for our problem.
φ
Lemma 1. Vector with φ=[b ;b ,...,b ,b ] is an eigenvector of the
(cid:18) 1 (cid:19) 0 1 N−1 N α
matrix M(λ).
The proof follows from the fact that φ is the fixed point of the fractional linear
transformation(13). Theproductofmatrices(18)correspondstotherepresentation
of s(φ) as a superposition s0◦s1◦···◦sN(φ), where s0(φ)=b0+φ,sk(φ)= λbk−+αφk
for k =1,2,...,N −1 and sN(φ)= λb∗N−+αNφ.
LetT(λ)= 1trM behalfofthetraceofthematrixM(λ),whichisapolynomial
2
ofdegreegorless. NotethatthedeterminantofM isequalto A= N (λ α )
− − i=1 − i
as it follows immediately from (18). Q
Lemma 2. The matrix (18) has the form
T(λ) B(λ) C(λ)
(20) M(λ)= − − ,
(cid:20) A(λ) T(λ)+B(λ) (cid:21)
where (A,B,C) is the α-triple of polynomials corresponding to φ. The discriminant
R=B2 AC equals to T2+A.
−
Indeed
P (λ) P (λ)
M(λ)= N−1 N ,
(cid:20) QN−1(λ) QN(λ) (cid:21)
where P , Q are defined above by (14). Now the first claim follows from the
k k
relations (17). Taking the determinant of both sides of (20) we have A = T2
B2+AC, which implies B2 AC =T2+A. − −
−
Now we need the following result about the factorisationof such matrices. This
kindofproblemsoftenappearsinthetheoryofdiscreteintegrablesystems(see[10]
and [11]).
PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES 7
Proposition 1. Let M(λ) be a polynomial matrix of the form (20), where A is a
monic polynomial of degree g, C is an anti-monic polynomial of degree g+1,T and
B arepolynomials ofdegreeg orless. AssumealsothatdetM(λ)= N (λ α ).
− i=1 − i
ThenforanopendensesetofsuchM thereexistsauniquefactorisatioQn oftheform
b λ α b λ α 1 b b
M(λ)= 0 − 1 ... N−1 − N N − 0 .
(cid:20) 1 0 (cid:21) (cid:20) 1 0 (cid:21)(cid:20) 0 1 (cid:21)
The proof is actually effective. We describe the procedure which allows to find
b uniquely assuming at the beginning that the factorisation exists.
i
Consider the transpose MT of the matrix M. For λ = α the matrix MT(λ) is
1
degenerate (since detMT(λ) = detM(λ) = N (λ α )). Find the null-vector
− i=1 − i
e = x1 of MT(α ), which is by definitionQany non-zero vector such that
1 (cid:18) y1 (cid:19) 1
(21) MT(α )e =0,
1 1
or explicitly
T(α ) B(α ) C(α ) x 0
1 − 1 1 1 = .
(cid:20) A(α1) T(α1)+B(α1) (cid:21)(cid:18) y1 (cid:19) (cid:18) 0 (cid:19)
It must satisfy the relation
b 1 x 0
0 1 =
(cid:20) 0 0 (cid:21)(cid:18) y1 (cid:19) (cid:18) 0 (cid:19)
sinceallotherfactorsarenon-degeneratewhenλ=α .Thisdeterminesb uniquely
1 0
as
T(α ) B(α ) C(α )
(22) b = 1 − 1 = 1 .
0 A(α ) T(α )+B(α )
1 1 1
−1
b λ α
Consider now the matrix M1 = (cid:20) 10 −0 (cid:21) M(λ). It is polynomial in λ
because of the following elementary
Lemma 3. Let M be the polynomial matrix, λ=α be a simple root of its determi-
−1
1 b λ α
nantande= beanullvectorofMT(α).Thenthematrix − M(λ)
(cid:18) b (cid:19) (cid:20) 1 0 (cid:21)
−
is polynomial.
X(λ) Y(λ)
Indeed, let M = then
(cid:20) Z(λ) W(λ) (cid:21)
b λ α −1 Z(λ) W(λ)
(cid:20) 1 −0 (cid:21) M(λ)=(cid:20) X(λ)−bZ(λ) Y(λ)−bW(λ) (cid:21).
λ−α λ−α
1
From MT(α) =0 it follows that λ=α is a root of the polynomials X(λ)
(cid:20) b (cid:21) −
−
bZ(λ)andY(λ) bW(λ).Thereforethesepolynomialsaredivisiblebyλ α,which
− −
proves the claim.
Repeat now the procedure by taking λ = α and so on. After N steps we will
2
come to a polynomial matrix
−1 −1
b λ α b λ α
MN(λ)=(cid:20) N1−1 −0 N (cid:21) ×···×(cid:20) 10 −0 1 (cid:21) M(λ)
8 M-P.GROSSETANDA.P.VESELOV
with determinant 1. To complete the proof of Proposition 1 we need to show that
1 b∗
M is of the form N .
N (cid:20) 0 1 (cid:21)
a λg+... λg+1+...
Recall that the matrix M(λ) is of the form 0 , where
(cid:20) λg+... d0λg +..., (cid:21)
the dotsmeantermsoflowerdegree,andthe coefficientsa andd maybe zero. It
0 0
−1 −1
b λ α b λ α
iseasytoshowthatthematrixM2(λ)=(cid:20) 11 −0 2 (cid:21) (cid:20) 10 −0 1 (cid:21) M(λ)
a λg−1+... λg +...
is of the form 2 and by induction M (λ) is of the
(cid:20) λg−1+... d2λg−1+... (cid:21) 2k
a λg−k+... λg−k+1+...
form k . Therefore the matrix M is of the form
(cid:20) λg−k+... dkλg−k+... (cid:21) N−1
−1
a λ+c b λ α
(cid:20) 1g dg (cid:21)whereag,c,dgareconstant. ThematrixMN =(cid:20) N1−1 −0 N (cid:21) MN−1
1 d
g
equals to . Since M is a polynomial matrix, we have
(cid:20) ag−bN−1 λ+c−bN−1dg (cid:21) N
λ−αN λ−αN
a =b and b d c=α . Thus M has the required form.
g N−1 N−1 g N N
−
We see that the procedure will not work only if at some stage the first com-
ponent of the null vector of MT(α ) vanishes. Clearly this happens only for a
k k+1
closedalgebraicsubsetofcodimension1,soforgenerictriples (A,B,C) the matrix
decomposition exists and is unique. This completes the proof of Proposition 1.
Now we are ready to finish the proof of Theorem 1 in the periodic case. Let
(A,B,C) be an α-triple, then by definition there exists a polynomial matrix S
of degree g or less such that the discriminant R = B2 AC is equal to S2 +A.
−
Clearly the polynomial S is defined up to a sign. Consider two corresponding
matrices M given by (20) with T(λ) = S(λ). Each of them generically has a
±
unique factorisation given by Proposition 1. One can easily check that this gives
two N-periodic α-fraction representations of the corresponding function φ(λ) =
−B(λ)+√R(λ)
and thus completes the proof in this case.
A(λ)
3. Pure periodic α-fractions
Letnowφ=[b ,b ,...,b ] beapureperiodicα-fraction. Thisisaparticular
0 1 N−1 α
case of the previous situation with b = b . But since the corresponding b∗ =
0 N N
b b =0, this case is actually degenerate and needs a special consideration.
0 N
−
First of all as before φ satisfies the relation
P φ+P
φ= N−1 N ,
Q φ+Q
N−1 N
where P , Q satisfy the relations (14), but now because b∗ =0 we have
k k N
P =(λ α )P , Q =(λ α )Q .
N N N−2 N N N−2
− −
Now from (17) we have
1
A(λ)=Q (λ), B(λ)= ((λ α )Q (λ) P (λ)), C(λ)= (λ α )P .
N−1 N N−2 N−1 N N−2
2 − − − −
Since Q and P are monic this shows that A,B,C satisfy the property (1)
N−1 N−2
of Theorem 1 with the additional condition C(α ) = 0. The proof of the second
N
property (α-admissibility of R) goes unchanged.
PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES 9
Now as in the previous case in order to find the pure periodic α-fraction one
should factorise the matrix
T(λ) B(λ) C(λ)
M(λ)= − −
(cid:20) A(λ) T(λ)+B(λ) (cid:21)
as the product
b λ α b λ α
0 − 1 ... N−1 − N .
(cid:20) 1 0 (cid:21) (cid:20) 1 0 (cid:21)
The main difference is that in the pure periodic case the trace T(λ) of the matrix
M(λ), which was known before only up to a sign, now is determined uniquely by
the condition
(23) T(α )= B(α ).
N N
−
Indeed T(α ) = 1(P (α ) + Q (α )) = 1(P (α )) = B(α ) since
N 2 N−1 N N N 2 N−1 N − N
Q (α )=0.If B(α )=0,which is a generic case,this determines T(λ)uniquely
N N N
6
(andthusthematrixM)byatriple(A,B,C).ThiscompletestheproofofTheorem
1.
Example. Consider the simplest case N = 1, g = 0. Then the α-triples have
the form
A=1, B =β, C = (λ+γ)
−
with arbitrary β, γ C, so that in the periodic case the corresponding φ have a
∈
general form
(24) φ= β+ λ+γ.
−
p
In this particular case this can be easily seen directly. Indeed φ=[b ,b ] satisfies
0 1 α
the quadratic equation
φ2+(b∗ b )φ (λ α +b b∗)=0,
1− 0 − − 1 0 1
where b∗ =b b . Thus to find a periodic continued α-fraction expansion of (24)
1 1− 0
one should solve the system of equations
(25) b∗ b =2β, b b∗ =α +γ,
1− 0 0 1 1
which has two solutions
b = β β2+α +γ, b∗ =β β2+α +γ.
0 − ± 1 1 ± 1
p p
One can easily check that these two solutions correspond to two solutions of the
factorisation problem from the previous section.
Inthepureperiodiccasebytheadditionalcondition(8)wehaveC = (λ α ),
1
− −
so γ = α and the general form of φ is
1
−
(26) φ= β+ λ α .
1
− −
p
In that case we have also b∗ = b b = 0, so the system (25) reduces to just one
1 1− 0
equation b = 2β, which determines the pure periodic α-fraction expansion of
0
−
(26) uniquely in agreement with our previous consideration.
10 M-P.GROSSETANDA.P.VESELOV
4. Action of Z S
2 N
×
A surprising corollary of Theorem 1 is the invariance of the set of N-periodic
α-fractions under the permutations σ S of the set α:
N
∈
σ(α) =α .
k σ(k)
This is not obvious from the very beginning and in fact is not true in the pure
periodic case.
In this sectionwe explainhow to use this symmetry to describe all 2N!periodic
α-fractions for a givenalgebraicfunction φ. In fact, the full symmetry groupis the
productG=Z S ,whereZ is generatedbythe involutionε interchangingtwo
2 N 2
⊕
different α-fraction expansions with the same order of the parameters α .
i
The action of this group is described by Theorem 2. We are going to prove it
now.
Recallthatπ S isthepermutation,whichreversestheorder1,2,...,N 1, N
N
∈ −
toN, N 1,...,2,1,andtheinvolutionσ swapsk andk+1leavingtherestfixed.
k
−
Let us startwith the actionof σ first. Let us introduce (assuming that b =0)
k k
6
α α α α
(27) ˜b =b + k+1− k, ˜b =b k+1− k, k =1,...,N 1.
k−1 k−1 b k+1 k+1− b −
k k
One can check directly the following matrix identity:
b λ α b λ α b λ α
k−1 − k k − k+1 k+1 − k+2 =
(cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)
˜b λ α b λ α ˜b λ α
k−1 − k+1 k − k k+1 − k+2 .
(cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)
b λ α b λ α 1 b b
Similarlyfork =N 1wehave N−2 − N−1 N−1 − N N − 0 =
− (cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)(cid:20) 0 1 (cid:21)
˜bN−2 λ−αN bN−1 λ−αN−1 1 ˜bN −b0 .Takingintoaccountthe
(cid:20) 1 0 (cid:21)(cid:20) 1 0 (cid:21)(cid:20) 0 1 (cid:21)
results of the previous section we see that the action of σ is indeed given by the
k
formula (9).
To provethe remaining partof Theorem2 recallthat φ=[b ;b ,...,b ,b ]
0 1 N−1 N α
isthe fixedpointofthe fractionallineartransformation(13),andthereforeitisthe
fixed point of its inverse, which as one can easily check is given by
a
s−1(φ)= b +b + N .
− N 0 b + aN−1
− N−1 −bN−2+...−b1+−ba01+φ
Thus
(28) φ=[b b ; b ,..., b , b ]
0− N − N−1 − 1 − N αN,...,α1,α0
is a periodic α-fraction corresponding to the sequence π(α)=α ,...,α ,α . Now
N 1 0
note that the trace of the corresponding matrix (19) is equal to P +Q =
N−1 N
(b +b + +b )λg+... (in the notations ofthe previoussection). Ifwe replace
1 2 N
···
b ,...,b by b ,..., b , b its highestcoefficientclearly changessign. This
1 n N−1 1 N
− − −
meansthatthenewperiodicα-fraction(28)correspondstotheactionoftheelement
επ G. Since επ and σ generate the group G we have described the full action.
k
∈
In the pure periodic case because of the additional condition C(α ) = 0 the
N
symmetry group is reduced to S , permuting α with i = 1,...,N 1. This
N−1 i
−