CC BY
5
УДК 515.17 + 517.545
Univalent Differentials of Integer Order on Variable Torus
Tatyana S. Krepizina*
Kemerovo State University, Red st., 6, Kemerovo, 650043, Russia
Received 06.05.2013, received in revised form 06.01.2014, accepted 20.06.2014 In this paper we give a full description for divisors of elementary differentials of all kinds. An analog of Appell's expansion formula for univalent functions on a variable torus is obtained. All basic type of vector bundles of meromorphic differentials of integer order over a Teichmuller space for torus are studied.
Keywords: univalent meromorphic differentials of integer order, divisors, vector bundles over Teichmuller space for torus.
Introduction
Univalent differentials (of order q =1 and q = 2 in particular) even on a fixed surface have found a lot of applications in mathematical physics (algebraic-geometric integration of nonlinear equations in the works of S.P. Novikov, I.M. Krichever), in theoretical physics (R. Dick), and also in analytic number theory in the works by H.M.Farkas and I.Kra [1].
The main difference of the results of this paper from the classical ones found in the books by J. Springer [2], H.M.Farkas and I.Kra [1] and in other books on the geometric function theory on a compact Riemann surface is that we consider all objects on a variable compact Riemann surface of genus g = 1 (torus) [3,4]. For the general theory of univalent differentials a big role is played by so called elementary differentials of integer order q that have the minimal number of poles: either one pole of order > 2, or two simple poles, and depend holomorphically on the modules of the torus For the first time we give a full description for divisors of elementary abelian q-differentials of all kinds. An analog of Appel's expansion formula for univalent functions on a variable torus is obtained. We study also all basic types of vector bundles of meromorphic differentials of integer order q =1 over a Teichmuller space for torus.
Preliminaries
Let F0 be a fixed compact Riemann surface of genus g = 1, F0 = C/r, where r is a group
with two generators Ai(z) = z + —, Bi(z) = z + — , Im — > 0. Let = — • The fundamental group of the surface Fo has an algebraic representation
r = ni(Fo) =< ai, bi : ai&i = biai > .
The class [F0, {ai, bi}] of conformally equivalent marked compact Riemann surfaces of genus one
is uniquely defined by a complex parameter (module) = —, which lies in the upper half plane
—
H = {z e C : Imz > 0}. Here F0 = C/r0 where r0 is the group generated by two generators
Aoi(z) = z +1, B0i(z) = z + ^0-
* kc-fabira@mail.ru © Siberian Federal University. All rights reserved
Every other class [FM, (a^, b^}] of conformally equivalent marked compact Riemann surfaces of genus one is uniquely defined by a complex parameter (module) ^ G H and FM = C/rM where rM is generated by AMi(z) = z +1, BMi(z) = z + Moreover, there is a quasiconformal mapping / : F0 ^ FM, and its lifting / : C ^ C on the universal covering surface gives an isomorphism between the marked group To and the marked group = /Mr0/—1 with a^ = f"M(a1), b^ =
U(bi).
The Teichmiiller space Ti = Ti(Fo), of the classes [FM, ( ai,b^ }] of conformally equivalent marked compact Riemann surfaces of genus one can be parametrized by points from H, and it is a 1-dimensional complex analytic manifold. This space with the Teichmuller metric is biholo-
morphically isometric to the space (H, ) , z = x + iy, with constant negative curvature [3].
V 2y /
Next, for every natural number n > 1 there is a fiber bundle over Ti such that its fibre over ^ G Ti as the space of all integer divisors of degree n on FM. Locally holomorphic sections of this bundle define on every FM an integer divisor DM of degree n that holomorphically depends on ^ [5, p.261,268].
Definition. A q-differential with respect to the group r on C is a differential ^(z)dzq such that
^(Tz)(T'z)q = ^(z), z G C, T G r.
In particular, for q = 0, this is a meromorphic function with respect to r. Let D be a divisor on F. Introduce following the spaces: L(D; F) of meromorphic functions / on F such that (/) ^ D, and (D; F) of meromorphic q-differentials w on F such that (w) ^ D. Denote by r(D) = dimC L(D; F) and iq(D) = dimC (D; F) the dimensions of these complex vector spaces.
Theorem (Riemann-Roch) [1, p. 73]. Let F be a compact Riemann surface of genus g = 1. Then for every divisor D on F
r(D_i) = deg D + i(D).
Theorem (Riemann-Roch for q-differentials) [4, p.43]. For every q G Z on a compact Riemann surface F of genus one
iq(D) = - deg D + r(1/D).
Theorem (Abel) [1, p.93; 4, p.67]. Let [F; (ai,bi}] be a marked compact Riemann surface of genus one and
Pai P am D = P ...Pm
Qi1 ...Qfs
be a divisor of degree zero on F. Then there exists a function / on F with
m s
(/) = D ^ f(D) = ^ ajf(Pj) - £ &f (Qk) = 0
j=i k=i
in J(F) = f (F), where f is the Jacobi mapping from F to J(F).
1. Univalent elementary q-differentials on a variable torus
In this section we establish the general form of elementary univalent q-differentials on the torus FM.
Let us find first the general form of q-differentials t^^q with the only pole Q = Q(^) exactly of order m ^ 2 on q G Z.
By the Riemann-Roch theorem for q-differentials on [4, p.43] we find the dimension
Qm) = dimc Qq(Qm; = - deg D + r(Qm).
where D = -. Hence iq (- | = m > 2. Here r(Qm) = 0, so deg(Qm) = m > 0 under our
Qm q Qm J v 7
conditions. This can also be proved by contradiction: if there existed a function g on such that (g) > Qm, then 0 = deg(g) > deg(Qm) > 2. Since deg Qm—1 = m - 1 > 1 > 0,
1 ^ - degf-m^ + r(Qm-1) = m - 1.
Qm—1 J & y Qm— 1 j
Therefore, iq | —^ | = iq ( ^ 1 , | + 1. Hence there exists a q-differential Tq"Q) with the pole
q V Qm / \Qm—V q'Q
exactly of order m at the point Q on , i. e. the divisor (t(!q) = —1 m m on —j =
Qm
Q, j = 1,..., m.
Construct now such a differential explicitly: = fdzq, q G Z, where dz is a holomorphic
differential on that depends holomorphically on The univalent function f has the divisor — • • • —
(f) = —Qm m, since (dz) = 1. By the Abel theorem [4] we get the equation
№ (M)(—1 ••• —m) - № (M)(Qm)=0
in the Jacobi manifold J(FM), where is an initial point different from Q. We understand this equation as an equality in the variable Jacobian J(FM), i. e. in the fibre of the universal Jacobi bundle that lies over the marked surface Therefore
y( —1) = ^(Qm) - ^(—2 ••• —m). (1)
Thus, for zeros of the function f we have m - 1 > 1 free parameters that can be arbitrarily chosen on locally holomorphically depending on By the theorem of C.Earle [5, p. 268] we can choose the divisor —2 • • • —m in such a way that it does not contain the point Q on and is a locally holomorphic section of the bundle of integer divisors of degree m - 1 over the Teichmiiller space T1.
Solving the Jacobi problem in the universal bundle over T1, we find the divisor —1 on which is a unique solution to the equation (1) [1, p. 95, 97]. Here the point —1 = Q and —1 depends holomorphically on our parameter, since the right hand side in (1) was chosen as holomorphically depending on Indeed, if —1 = Q then consider the divisor D = —2...—m with m - 1 free points. By the theorem on free points [1, p. 125] we have the inequality
m - 1 + 1 ^= m - 1 + i(D),
and hence 1 < i(D). Therefore we see that there exists a differential w = 0, (w) > D. Consequently, we have an impossible inequality
0 = deg(w) ^ deg D = m - 1 ^ 1. Thus, the divisor (T(rQ) = —1— 2 m —m is the most general for q-differentials T(rQ with the
Qm
only pole exactly of order m > 2 on with the point Q G Therefore, we have proved the following theorem.
Theorem 1.1. On a variable torus for every natural number m > 1, q £ Z there exists an elementary q-differential t^q with the pole at the point Q = Q(^) £ exactly of order m locally holomorphically depending on whose divisor is of the form
f (m)\ = Rl • • • Rm
Vq'Q) = Qm '
where
^(Rl)= ^(Qm) - ^(R2 ••• Rm). Here the divisors R2...Rm and Q = Q(^) are chosen as locally holomorphic sections of the bundle of integer divisors over T1 of degrees m — 1 and 1 respectively for ^ from a sufficiently small neighborhood U(^,0) C T1.
Corollary 1.1. Under the assumptions of theorem 1.1 there exists a q-differential
1
. Z
in a neighborhood of the point Q on
Proof. For every q £ Z, m > 1, there exists a q-differential
^(m) _ / c_m c_1
z"
t (m)
1 a-Q
in a neighborhood of the point Q on . The Abelian 1-differential ^ ^ has the residue c_1 = 0 at the point Q by the residue theorem on For m = 2 we have a q-differential
_1_ T (2) = (1
c_2 Ta-Q = ^ z2 By induction, for every m > 1 we can get a q-differential
=( ^ + O(1)) dzq
(C" + - + ^ + O(l0 dz', C-m =0
= ^ ^ = ( z2 + °(i)) dzq •
= ( im + o(in dzq
'q;Q
z"
in a neighborhood of the point Q on Moreover, such q-differential can be obtained by differentiating with respect to the parameter z(Q) from the formula
~(m) = 1 r~(2) ]("-2)
q;Q (-m + 1)...(-2) L q;QQ •
Thus, we have proved the corollary. □
Remark 1.1. For every q £ Z by the Riemann-Roch theorem for q-differentials we have the equality
iq (D) = - deg D + r ( D
and ia(1) = 1. Therefore ia ^= 1 + r(Q) = 1. Also ia ^= r = 1, where the first
equality follows from the isomorphism given by division by the differential dza on the torus
Because of that we have ia ^= 1 = ia(1). Therefore, there is no a q-differential Ta;Q on the
torus with the only pole at Q exactly of order one for every q £ Z. This fact can be also proved by using the residue theorem for abelian differentials of order one on [7, 8].
Now we establish the general form for univalent q-differentials Ta;Q1Q2 of the third kind with exactly two simple poles at different points Q1 = Q1(^) and Q2 = Q2(^) on that depend holomorphically on the parameter
Proposition 1.1. On a variable torus for every integer q there exists an elementary q-differential t9;q1q2 of the third kind with exactly two simple poles at different points Qi = Qi(y) and Q2 = Q2(«) on locally holomorphically depending on y with the divisor (t9;q1q2 ) =
RiR2
———, where y(Ri) = ¥>(QiQ2)-<£>№) in J(FM). Here the points R2, Qi = Qi(y), Q2 = Q2(y) Q1Q2
can be chosen as locally holomorphic sections of the bundle of integer divisors of degree one over Ti for y from a sufficiently small neighborhood U(y0) C Ti.
Proof. For q G Z, set t9;q1q2 = tq1q2dzq-:L, where tq1q2 is the classical abelian differential of the third kind on that depends holomorphically on y [1, p.51; 6].
Such a differential t9;q1q2 can also be taken as t = fdzq, where f is a univalent function Ri R2
with the divisor (f ) = . By Abel's theorem we have the equality
Qi Q2
^(Ri) = ^(QiQ2) - ^(R2) (2)
in J(FM). The divisor Ri is the only solution to the equation (2). Moreover, we can take the points such that Rj = Qi, Q2, j = 1,2. Indeed, if Ri = Qi for R2 = Qi, Q2, then <^(R2) = ^(Q2) and R2 = Q2. We arrive at a contradiction which proves the preposition. □
2. An analog of Appell's expansion formula for meromorphic functions on a variable torus
In this section we find an analog of Appell's formula where the terms (summands) have poles only at one point on and depend holomorphically on y.
Let f be a function on a variable torus with s simple poles Qi,Q2, ...,Qs and residues ci, ...,cs at them respectively. Consider the expression
fi = f - ciïQ? -... - CsT^
where = -/tS is a branch of the _y abeQn in.gra, f the second kH
[1, p.51] with only simple pole at Qk and the residue +1 at Qk depends holomorphically on y, k = 1,..., s. Then fi is an abelian integral of the first kind on the torus Therefore
fi = C y dz + C = Ciz + C on
Theorem 2.1. Let f be a function on a variable torus with simple poles Qi,...,Ql and residues ci,...,cl at them, and poles at Ql+i ,...,Qs with multiplicities nl+i,..., ns, nk ^ 2, k = l + 1,..., s, and given principal parts at them. Then
1
f = Ciz + C + £ cj j
j=i
+ E
fc=i+i
dT(1) A d2T(1) zl d"fc "IT(1)
r(1) + A Qk + Afc,3 d T Q fc + + Afc,nfc d TQ k
Qk + dQk + 2! dQk + - +(nfc - 1)! dQ?k-1
where Ci,C are complex numbers and
f =_^ink_+ +_^_+ Afc-1 + O(1)
7 (z - z(Qfc))"^ " +(z - z(Qfc))2 + z - z(Qfc
in a punctured neighborhood of k = l + 1,..., s, on and all terms depend holomorphically on
Proof. If Q1 is a pole of order n1,n1 > 2, then in the previous formula the term c1TQ1i) is replaced by the sum
m dTQ1) d2TQ1) 5"i_1TQ1)
A rT(1) i A Qi i 13 Qi i i Q1
Qi + + ^ ^OT + ... + (n1 — 1)! SQ--! '
where are the coefficients of the principal part of the Laurent series for the function f in a punctured neighborhood of the point Qk= 1, ...,nk(Qk), k = l + 1,..., s. Indeed, in a
neighborhood of the point we have the expansions tQ^ = -' + O(1),z(Qk) = ak;
Wk z — z(Qfc)
(TS)afc = (T—^ + O(1);...; (T^)Й° = (z - m!)m+1 + oa), 1 < m < nfc(Qfc) - 1> where nk (Qk) is the order of the pole at the point for f> k = 1 + 1,..., s. The theorem is proved. □
3. The space of meromorphic q-differentials on a variable torus
Denote by (-501-DaL-5" ; F^J the space of q-differentials on FM that are mul-
VP1 ••• pi pi+1 ••• pn /
tiples of the divisor —a- -—, where q G Z> a1>...>« > 2, n > 1, 0 < 1 < n>
P1 ••• pi pi+1 ••• Pn
and the points P^ ...>Pn are pairwise distinct, and by (1; FM) the subspace of holomorphic q-differentials on FM. The divisor pa • • • P^1 Pl+1 • • • Pn is chosen as a locally holomorphic section of the bundle of integer divisors of degree a1 + • • • + аг + n — 1 over T1.
By the Riemann-Roch theorem for q-differentials we find the dimensions of these spaces. For every q we have dirnQ9(1; FM) = 1, and
1 Л = — deS f ЪО-^-5-) + r (p1ai • • • рга1 рг+1 • • • pn) =
1
q\ pa ••• PaРг+1 ••• P
г 'Pi+1 ••• Pn J VPi ••• p P+1
= «1 +----+ « + n — 1 (> 1).
Therefore
dimQ^Pa ^ pJPi+1 ^ Pn ; F^j (1; FM) = «1 + • • • + a + n — 1 — 1 (> 1).
Consider the following collections of q-differentials:
_(2) _(«i) _(2) _(«¡) _ _ for i i; (4)
Tq;PiP2> ... >Tq;PiPn> for 1 = °. (5)
Let us show that the coset classes of q-differentials from (4) are linearly independent over C. Assume that there exists a linear combination of differentials from (4)
C(2) T(2) + + C(ai)T (a ) + + C(2) (2) + + C(a ) (аг) + + + = ;
C1 Tq;Pi +-----+ C1 Tq;Pi +-----+ Cl Tq;Pi +-----+ Cl Tq;Pi + C2Tq;Pi P2 +-----+ CnTq;PiP„ =
where w is a holomorphic q-differential, such that not all its coefficients are zeroes.
The coefficients C(2) = • • • = C(ai) = 0, since in the right hand side the points Pi, ...,P; are not poles of order > 2. We are left with the equality
C2Tq;PiP2 + • • • + Tq;PlPn =
Since the points P2, Pn are not singular for the right hand side, C2 = • • • = Cn = 0. Thus, the coset classes for q-differentials from (4) is a base for the quotient space.
Let us now show that the collection (5) is linearly independent over C. Suppose that there exists a linear combination O2Tq;PiP2 + • • •+OnTq;PiPn = w, where w is a holomorphic q-differential, such that not all its coefficients are zeroes. The coefficients C2 = • • • = Cn =0, since P2,..., Pn are not singular for the right hand side. Therefore the coset classes of q-differentials from (5) form a base for the quotient space. Thus, we have proved the following theorem.
Theorem 3.1. The vector bundle E = |J Qq ( --—; fA /Qq(1; of rank
M VPi • "pn /
d = a1 + • • • + a, + n — l — 1, where a1, ..., a, ^ 2, n ^ 1, 0 ^ l ^ n, q G Z, over T1 is complex analytic equivalent to the direct product T1 x Cd, and the coset classes of q-differentials from the collections (4), (5) give a base of locally holomorphic sections of this bundle over T1.
Consider the collection of q-differentials
7 q (2) (ai) (2) (a)
dz ; Tq;Pi, ..., Tq;Pi , Tq;P , Tq;P , Tq; Pi P2 , Tq; Pi P„ . (6)
Let us show that q-differentials from(6) are linearly independent over C. Assume again that there exists a linear combination
Cdzq + C(2)T(2) + + C (ai)T (ai) + O(2)T(2) + + C(a)T(a) +
°1dz + O1 Tq;Pi + r C1 Tq.Pi + Tq^ + + Tq^ +
+ C2Tq;PiP2 +-----+ On Tq.Pi Pn = 0,
such that not all its coefficients are zeroes. The coefficients C(2) = • • • = C,(ai) = 0 and C2 = • • • = Cn =0, since in the right hand side there are no singular points. So we have O1dzq = 0, which implies O1 = 0. Therefore the collection (6) of q—differentials is a base for the space 1
Qq —ai- a -—; Fm . This prove the following theorem.
VP1 •••p p;+1 • "pn J
Theorem 3.2. The vector bundle E2 = J Qq ( -5a-po^n-5-; Fm) of rank d1 = a1 +
M VP1 • Pi+1^ --Pn / • • • + a, + n — l over T1 is complex analytic equivalent to the direct product T1 x Cdi. Moreover, q-differentials of (6) give a base of locally holomorphic sections of this bundle over T1, where a1, ..., a, ^ 2, n ^ 1, 0 ^ l ^ n, and q G Z.
Remark 3.1. For q = 0, l = 0 the collection (6) is 1,To;PiP2 = f2,..., T0;PiPn = fn and is a base for the space ^{^p ^ p ;Fm^ , where fj are non-constant functions, and (fj) > p~pj, j = 2,..., n, on Fm.
Remark 3.2. In particular, for q =1 and for a fixed torus F Corollary 1.1 and Theorems 2.1, 3.1, 3.2 imply classical theorems on abelian 1-differentials found in [1,2].
References
[1] H.M.Farkas, I.Kra, Riemann surfaces, New-York, Springer, 1992.
[2] G.Springer, Introduction to Riemann Surfaces, Addison-Wesley, Massachusetts, 1957.
[3] L.V.Ahlfors, L.Bers, Spaces of Riemann surfaces and quasi-conformal mappings, Moscow, 1961 (in Russian).
[4] V.V.Chueshev, Multiplicative functions and Prym differentials on variable compact Riemann surface, Part 2, Kemerovo, 2003 (in Russian).
[5] C.J.Earle, Families of Riemann surfaces and Jacobi varieties, Annals of Mathematics, 107(1978), 255-286.
[6] V.N.Monahov, E.V.Semenko, Boundary problems and pseudodifferential operators on Riemann surfaces, Moscow, FIZMATLIT, 2003 (in Russian).
[7] T.S.Krepizina, Divisors of Prym differentials and Abelian differential on torus, Vestnik KemGU, 1(2011), no. 3, 206-211 (in Russian).
[8] T.S.Krepizina, V.V.Chueshev, Multiplicative functions and Prym differentials on variable tori, Vestnik NGU, 12(2012), no. 1, 74-90 (in Russian).
Однозначные дифференциалы целого порядка на переменном торе
Татьяна С. Крепицина
В этой 'работе дано полное описание дивизоров элементарных дифференциалов всех 'родов. Получен аналог формулы Аппеля разложения однозначной функции на переменном торе. Исследованы основные типы векторных расслоений из мероморфных дифференциалов целого порядка над пространством Тейхмюллера для тора.
Ключевые слова: однозначные мероморфные дифференциалы целого порядка, дивизоры, векторные расслоения, пространство Тейхмюллера.
