# 电竞俱乐部解散: Periodic continued fractions and hyperelliptic curves

arXiv:math/0701932v1 [math.GM] 31 Jan 2007 PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES M-P. GROSSET AND A.P. VESELOV Abstract. We investigate when an algebraic function of the form φ(λ) = ?B(λ)+√R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coeffi cients in C, can be written as a periodic α-fraction of the form φ(λ) = [b0;b1,b2,.,bN]α= b0+ λ ? α1 b1+ λ?α2 b2+. b N?1+ λ?αN bN+ λ?α1 b1+ λ?α2 b2+. , for some fi xed sequence αi. We show that this problem has a natural answer given by the classical theory of hyperelliptic curves and their Jacobi varieties. We also consider pure periodic α-fraction expansions corresponding to the special case when bN= b0. 1. Introduction Consider the following continued fraction, which we will call α-fractions: (1)φ = b0+ λ ? α1 b1+ λ?α2 b2+ . = [b0,b1,.,]α, where α = (αi),αi∈ C is a given sequence, biare arbitrary complex numbers, λ is a formal parameter. In this paper we will consider a special case of N-periodic α-fractions, when the sequences αiand biare periodic with period N : αi+N= αi, bi+N= bi for all i ≥ 1 : (2)φ = [b0;b1,b2,.,bN]α. In the particular case when bN= b0we have φ = [b0,b1,.,bN?1]α, which will be called a pure N-periodic α-fraction. This kind of fractions naturally appear in the theory of integrable systems, in particular in the theory of periodic dressing chain [1], but to the best of our knowl- edge has not been studied so far. We were partly inspired by our recent discussions with Vassilis Papageorgiou on the discrete KdV equation where such continued fractions appear as well [2]. 1 2M-P. GROSSET AND A.P. VESELOV Because of periodicity we can write formally (2) as φ = b0+ λ ? α1 b1+ λ?α2 b2+ .bN?1+ λ?αN bN?b0+φ , which implies a quadratic relation (3)A(λ)φ2+ 2B(λ)φ + C(λ) = 0, where A,B,C are certain polynomials in λ with coeffi cients polynomially depending on bi. Thus to any periodic α-fraction (2) corresponds an algebraic function (4)φ(λ) = ?B(λ) + pR(λ) A(λ) , 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 expansion of the algebraicfunction (4) from the hyperelliptic extension C(λ, pR(λ)) of the fi eld of rational functions C(λ). We leave the question of convergence aside concentrating on algebraic and geometric aspects of the problem. We will discuss the following three main questions. Question 1. Which algebraic functions (4) admit N-periodic α-fraction expan- sions ? Question 2. How many such expansions may exist for a given algebraic function (4) and how to fi nd them ? Question 3. What is the geometry of the set of functions (4) from given hyper- elliptic extension (i.e. with fi xed 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 αiare distinct. 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 also that in the classical number-theoretic version the 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 fraction expansion ξ = [a0,.,ak] if and only if ξ is larger than 1 and its conjugate ˉ ξ = p ? q√d lies between ?1 and 0 (see e.g. [4]). Note also that the periodic continued fractions of the form [a0;a1,.,ak] also appear naturally in number theory as expansions of √d (see [4]). To explain our main results let us introduce the polynomial (6)A(λ) = N Y i=1 (λ ? αi) and call a polynomial R of degree N α-admissible if (7)R(λ) = S2(λ) + A(λ) PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES3 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 coeffi cient 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) deg B ≤ 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 bi are rational functions of both coeffi cients of A, B, C and parameters αiand can be found by an eff ective matrix factorisation procedure. In the pure periodic case the only additional requirement is (8)C(αN) = 0, under which the pure periodic α-fraction expansion is generically unique. We will call (A,B,C) satisfying the conditions (1), (2) of Theorem 1 the α-triples. Note that these conditions are invariant under any permutation of the parameters αi. In fact there is a natural birational action of the direct product G = Z2× SN on the set of periodic continued α-fraction expansions, where the generator ε of Z2 is acting simply by swapping two diff erent α-fraction expansions given by Theorem 1. Our next result describes this action explicitly. Let us introduce the following permutation π ∈ SN, which reverses the order α1,α2,.,αN?1, αNto αN, αN?1,.,α2,α1and the involutions σkswapping αk and αk+1, where k = 1,.,N ? 1. We will show that σkis acting on b = (bi), i = 0,.,N with bk6= 0 as follows: (9) ? bk?1= bk?1+ αk+1? αk bk , ? bk+1= bk+1? αk+1? αk bk , the rest of biremain the same. This determines the action of the symmetric group SNsince σkgenerate it. To describe the action of Z2it is enough to describe the action of the involution επ ∈ G, which turns out to be quite simple: (10) ? bj= ?bN?j, j = 1,.,N ? 1 ? b0= b0? bN, ? bN= ?bN. Theorem 2. The formulae (9) and (10) defi ne a birational action of the group G = Z2× SNon the set of N-periodic α-fractions. Its orbits consist of all 2N! possible periodic α-fraction expansions for a given α-triple (A,B,C) and any permutation of the parameters αi. In the pure periodic case the symmetry group is broken down to SN?1generated by σkwith k = 1,.,N ? 2 given by (9). Let us fi x now the α-admissible polynomial R(λ) with distinct roots. We would like to describe the geometry of the set of elements from the hyperelliptic extension fi eld C(λ, pR(λ)) which have a pure periodic α-fraction expansion. For the stan- dard material from the algebraic geometry of the curves we refer to the classical Griffi ths-Harris book [5]. Consider the hyperelliptic curve ΓRgiven by the equation (11)μ2= R(λ). 4M-P. GROSSET AND A.P. VESELOV The curve ΓR consists of the affi ne part Γaff R , corresponding to the ”fi nite” solutions of (11), and the ”infi nity” point, which we will denote as P∞. Since the roots of R are distinct it is non-singular and has genus g. Consider g points P1,.,Pgof Γaff R and call the corresponding divisor D = P1+ ··· + Pgnon-special if (12)Pi6= τ(Pj) for any i 6= j, where τ is the hyperelliptic involution: τ(μ,λ) = (?μ,λ). Non-special divisors have the property that the linear space L(D) of all meromor- phic functions on ΓRhaving the poles at D of order less than or equal to 1 has 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 infi nity. These functions are in a way ”least singular” among the ”generic” meromorphic functions on ΓRin the sense that any such function can not have less than g + 1 poles (see [5]). Now defi ne the affi ne Jacobi variety J(ΓR)affas the set of positive non-special divisors D = P1+ ··· + Pg, P1,.,Pg∈ Γaff R . Let Mα R be an affi ne variety of α-triples of polynomials (A, B, C) with given discriminant R(λ), and Pα R be its subvariety given by the additional condition (8). Theorem 3. There exists a bijection between the set Mα R and the extended affi ne Jacobi variety J(ΓR)aff×C. The corresponding algebraic functions (4) can be char- acterised as meromorphic functions φ ∈ L(D + P∞) on ΓRwith non-special pole divisor D + P∞and asymptotic φ ～ √ λ at infi nity. In the pure periodic case under the assumption that R(αN) 6= 0 there exists a natural 2 : 1 map from the set Pα R to J(ΓR)aff. The corresponding φ from L(D + P∞ ) are fi xed by the condition that one of two values of φ(αN) is zero. The proof is based on the classical description of the Jacobi variety due to Jacobi himself [6] (see also Mumford [7]). We see that the affi ne space CNof all functions (4) having periodic α-fraction expansion is birationally equivalent to the double covering of the bundle of the extended affi ne Jacobians of α-admissible hyperelliptic curves. This is a version of the well-known result by Dubrovin and Novikov [8] who were the fi rst 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 φ = b0+ λ ? α1 b1+ λ?α2 b2+ .bN?1+ λ?αN bN+ λ?α1 b1+ λ?α2 b2+. = [b0;b1,.,bN?1,bN]α PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES5 We see that (at least formally) φ is the fi xed point of the fractional linear trans- formation (13)s(φ) = b0+ a1 b1+ a2 b2+ .bN?1+ aN b? N+φ with b? N = bN? b0and ak= λ ? αk, k = 1,,N. The function s(φ) can be written as s(φ) = PN?1φ+PN QN?1φ+QN, where the quantities Pk, Qk are determined by the standard recurrence relations (see e.g. [9], page 14): P?1= 1,Q?1= 0, P0= b0,Q0= 1, (14)Pk+1= bk+1Pk+ ak+1Pk?1, Qk+1= bk+1Qk+ ak+1Qk?1 for k N ? 1 and PN= b? NPN?1+ aNPN?2, QN= b? NQN?1+ aNQN?2. Thus we have (15)φ = PN?1φ + PN QN?1φ + QN , which can be written as a quadratic equation (16)QN?1φ2+ (QN? PN?1)φ ? PN= 0. It is easy to see from the recurrence relations that Pkand Qkare polynomials in λ of the form P2k= (b0+ b2+ . + b2k)λk+ .,P2k+1= λk+1+ ., Q2k= λk+ .,Q2k?1= (b1+ b3+ . + b2k?1)λk?1+ ., for k ≤ g and Q2g+1= (b1+ b3+ . + b2g?1+ b2g+1? b0)λg+ ., where the dots denote the lower degree terms. Since from (16) (17)A(λ) = QN?1(λ),B(λ) = 1 2(QN(λ) ? PN?1(λ)), C(λ) = ?PN(λ), 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 β = ?b0+ 1 2 N X k=1 (?1)k+1bk. Let us show now that the discriminant R = B2? AC is α-admissible. We have R = 1 4(QN ? PN?1)2+ PNQN?1= 1 4(PN?1 + QN)2+ PNQN?1? PN?1QN. We claim that PNQN?1? PN?1QN= N Y i=1 (λ ? αi). Indeed, the determinant ? ? ? ? PNPN?1 QNQN?1 ? ? ? ? = ? ? ? ? bNPN?1+ aNPN?2PN?1 bNQN?1+ aNQN?2QN?1 ? ? ? ? = 6M-P. GROSSET AND A.P. VESELOV ?aN ? ? ? ? PN?1PN?2 QN?1QN?2 ? ? ? ? = ··· = (?1)NaNaN?1.a1 ? ? ? ? b01 10 ? ? ? ?. Since N is odd, ? ? ? ? PNPN?1 QNQN?1 ? ? ? ? = aNaN?1.a1= N Y i=1 (λ ? αi) = A. Now by taking S(λ) = PN?1+QN 2 , which is a polynomial of degree g or less, we see that R(λ) = S2 +A, so R is α-admissible. This proves the fi rst part of Theorem 1 in the periodic case. To prove the second part let us introduce the following matrix (18)M(λ) = ? 1b0 01 ?? 0λ ? α1 1b1 ? . ? 0λ ? αN 1b? N ? , with b? N = bN? b0. One can check that it can be rewritten also as (19)M = ? b0λ ? α1 10 ? . ? bN?1λ ? αN 10 ?? 1b? N 01 ? . The following Lemma explains its importance for our problem. Lemma 1. Vector ? φ 1 ? with φ = [b0;b1,.,bN?1,bN]αis an eigenvector of the matrix M(λ). The proof follows from the fact that φ is the fi xed point of the fractional linear transformation(13). The product of matrices (18) corresponds to the representation of s(φ) as a superposition s0?s1? ···? sN(φ), where s0(φ) = b0+ φ,sk(φ) = λ?αk bk+φ for k = 1,2,.,N ? 1 and sN(φ) = λ?αN b? N+φ . Let T(λ) = 1 2trM be half of the trace of the matrix M(λ), which is a polynomial of degree g or less. Note that the determinant of M is equal to ?A = ? QN i=1(λ?αi) as it follows immediately from (18). Lemma 2. The matrix (18) has the form (20)M(λ) = ? T(λ) ? B(λ)?C(λ) A(λ)T(λ) + B(λ) ? , where (A,B,C) is the α-triple of polynomials corresponding to φ. The discriminant R = B2? AC equals to T2+ A. Indeed M(λ) = ? PN?1(λ)PN(λ) QN?1(λ)QN(λ) ? , where Pk, Qk are defi ned above by (14). Now the fi rst claim follows from the 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 factorisation of such matrices. This kind of problems often appears in the theory of discrete integrable systems (see [10] and [11]). PERIODIC CONTINUED FRACTIONS AND HYPERELLIPTIC CURVES7 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 are polynomials of degree g or less. Assume also that detM(λ) = ? QN i=1(λ?αi). Then for an open dense set of such M there exists a unique factorisation of the form M(λ) = ? b0λ ? α1 10 ? . ? bN?1λ ? αN 10 ?? 1bN? b0 01 ? . The proof is actually eff ective. We describe the procedure which allows to fi nd biuniquely assuming at the beginning that the factorisation exists. Consider the transpose MTof the matrix M. For λ = α1the matrix MT(λ) is degenerate (since detMT(λ) = detM(λ) = ? QN i=1(λ ? αi)). Find the null-vector e1= ? x1 y1 ? of MT(α1 ), which is by defi nition any non-zero vector such that (21)MT(α1)e1= 0, or explicitly ? T(α1) ? B(α1)C(α1) A(α1)T(α1) + B(α1) ?? x1 y1 ? = ? 0 0 ? . It must satisfy the relation? b01 00 ?? x1 y1 ? = ? 0 0 ? since all other factors are non-degenerate when λ = α1. This determines b0uniquely as (22)b0= T(α1) ? B(α1) A(α1) = C(α1) T(α1) + B(α1). Consider now the matrix M1= ? b0λ ? α 10 ??1 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- nant and e = ? 1 ?b ? be a null vector of MT(α). Then the matrix ? bλ ? α 10 ??1 M(λ) is polynomial. Indeed, let M = ? X(λ)Y (λ) Z(λ)W(λ) ? then ? bλ ? α 10 ??1 M(λ) = ? Z(λ)W(λ) X(λ)?bZ(λ) λ?α Y (λ)?bW(λ) λ?α ? . From MT(α) ? 1 ?b ? = 0 it follows that λ = α is a root of the polynomials X(λ) ? bZ(λ) and Y (λ)?bW(λ). Therefore these polynomials are divisible by λ?α, which proves the claim. Repeat now the procedure by taking λ = α2and so on. After N steps we will come to a polynomial matrix MN(λ) = ? bN?1λ ? αN 10 ??1 × ··· × ? b0λ ? α1 10 ??1 M(λ) 8M-P. GROSSET AND A.P. VESELOV with determinant 1. To complete the proof of Proposition 1 we need to show that MNis of the form ? 1b? N 01 ? . Recall that the matrix M(λ) is of the form ? a0λg+ .λg+1+ . λg+ .d0λg+ ., ? , where the dots mean terms of lower de