CC BY
70DOI: 10.15393/j3.art.2018.5290
UDC 517.545
S. R. NASYROV
FAMILIES OF ELLIPTIC FUNCTIONS AND UNIFORMIZATION OF COMPLEX TORI WITH A UNIQUE POINT OVER INFINITY
Abstract. We investigate the problem of describing a one-parametric family of elliptic functions which uniformizes a given family of ramified coverings of the Riemann sphere with maximal possible ramification over infinity. We find a PDE for the family of functions and use it to deduce a system of ODEs for their critical points.
Key words: Riemann surface, complex torus, elliptic function, uniformization
2010 Mathematical Subject Classification: 30F10, 32G15
1. Introduction. Let S be an n-sheeted compact Riemann surface over the Riemann sphere C with a unique point over infinity of multiplicity n — 1. We will mainly consider the case when the genus p of S equals 1, i.e., S is a complex torus; however, in Section 2, following [12], we also shortly describe the simply-connected case (p = 0).
If p = 0, then, by the well-known Koebe-Poincare uniformization theorem, there exists a polynomial f which maps C onto S conformally. In the case p =1 the surface S can be uniformized by an elliptic function, i.e., there exists a two-periodic meromorphic function f with periods and such that the factor-mapping f : C/Q ^ S is a conformal isomorphism. Here Q is the lattice generated by and u2.
Denote the critical points of f by aj, 1 ^ j ^ N. Since S is given, we know the critical values Aj = f (aj), 1 ^ j ^ N, and their multiplicities mj. The problem of uniformization of S is reduced, in essence, to finding the values aj, 1 ^ j ^ N.
The system
f (aj ) = Aj, 1 ^ j ^ N, (1)
©Petrozavodsk State University, 2018
is a system of non-linear equations with respect to aj, with certain difficulties in solving. Hurwitz [3] first noted that, critical values are not defined uniquely by given Aj and multiplicities mj. He posed the problem of the number of non-equivalent coverings with the same branch data.
(k)
The Hurwitz problem. Let p G Z+, naturals numbers nj ), j = 1,... ,hk, k = 0,...,m, satisfy
hk
£• *(k)
j=i
m hk
nj ) = n, k = 0,... ,m,
££(j - 1) = (2 - 2p)n - 2, k=0 j=l
and a finite set T = {T0, Tl,..., Tm} C C be given. Find how many non-equivalent compact Riemann surfaces over C of fixed genus p exist, such that their branch points lie over T and for every k there exist exactly hk points of multiplicities nlk) — 1,..., nhk) — 1, lying over Tk, k = 0,..., m.
The Hurwitz problem and its generalizations were investigated, in particular, in [3-10,14]. The obtained results show that (1) can have multiple solutions, and distinct solutions can correspond to non-equivalent coverings of the Riemann sphere. Therefore, even if we solve system (1) and find several solutions, we have to choose the one that matches the given surface S.
In the paper we propose an approximate method of finding aj. Its essence is in the following. Consider the space £ of all n-sheeted Riemann surfaces of genus p, which are ramified covering of C, with the same branch data as S. The topology on £ is induced by the Caratheodory kernel convergence (see, e.g., [10]). Let us include S into a smooth one-parametric family S(t), 0 ^ t ^ 1, such that S(1) = S, and the uniformizing function f (z, t) for S(t) is known for t = 0; therefore, we know its critical points a0. Smoothness of the family means that projections Aj (t) of branch points depend smoothly on t.
We describe the movement of aj (t) with the help of a system of ODEs (Theorem 7). Solving the Cauchy problem for the system with the initial data (a0) we obtain dependences aj (t), 0 ^ t ^ 1, and the final values (t =1) give the desired critical values aj for S. Note that in the case p = 1 we need to determine, additionally, the parameter /wl that depends on t, called the module of the complex torus S.
To obtain the system of ODEs mentioned above, we deduce the PDE
fM (2)
which the family f (z,t) satisfies to (Theorem 6). The idea of using one-parametric families of analytic functions and equations of such type belongs to Loewner who offered his famous parametric method in the theory of univalent functions; a more general equation of the form (2) was suggested by Kufarev (see, e.g., [2]).
Note that in [13] we investigated one-parametric families of elliptic functions with simple critical points. Here we study the case of arbitrary multiplicities.
2. The simply-connected case. The case p = 0 is described in [11]. In [12] a more general case is investigated: when f (z,t) is a family of rational functions. Here we formulate the main results from [12] specifying them for the case of polynomials.
Consider a smooth one-parametric family of polynomials
-1 7n N-1 n-1
P (z,t) = n (Z - aj )m-1dZ = - - £ mj aj ^ + ..., (3)
a i
n ^ " 3 3 n - 1
3 = 1 3 = 1
0 ^ t ^ 1, where aj = aj (t) are pairwise distinct points of the complex
plane, N ^ 3, mj — 1 E N, J^j 11 (mj — 1) = n — 1. We will also assume that
w-1
m3 a3 = 0
This condition can be fulfilled by a shift of the z-plane. Denote
H (z ) =-sir1-.
n (z - aj)m
3 = 1,3=1
Let Pj,m -1(z,t) be the Taylor's polynomial of of degree (mj — 1)
at the point aj and Aj (t) = f (aj (t), t), 1 ^ j ^ N — 1. From (3) it follows that A1 (t) = 0. This does not restrict generality because the condition can be provided by a shift in the plane.
Theorem 1. The family P(z,t) satisfies the PDE
P(z,t) = h(z,t)P'(z,t), z G C, 0 ^ t ^ 1,
where
h(z,t) = V "I(z,t) Aj(t).
j=2 (z - aj(t))m j()
Here and further the dot above a letter denotes differentiation with respect to t (the partial derivative if function depends on z and t, and the ordinary one for function of the only variable t), and the prime means the derivative by z.
Theorem 2. The critical points aj (t) satisfy the system of ODEs
a, = Hm'-l)(a;) A, + V GmrilMAk, 1 S l S N - 1,
(m< - 1)! 1 k=ti=, <mk - 2)!
where Gk,(z) = Hk(x)/(z - a,).
Note that (4) allows to find the critical points of the polynomial, uni-formizing a given simply-connected Riemann surface, quickly and with a good accuracy. While solving (4), we obtain the whole family of polynomials uniformizing a definite family of surfaces. If we know a solution for a set of Aj with fixed branch data, then we can obtain other solutions moving Aj along various closed curves.
We can also consider (4) as variational formulae for polynomials, expressing variations of critical points through variations of critical values. The formulae can be useful for solving some extremal problems of geometric function theory connected with polynomials.
3. Preliminary results on elliptic functions. Now we will investigate smooth one-parametric families of n-sheeted complex tori. We assume that for every torus there exists a unique point lying over infinity and connecting all its sheets. The case when all other branch points are simple was studied in [13]; here we investigate the case of arbitrary multiplicities.
First we recall some classical results of the theory of elliptic functions (se^ e.g^ [1]).
A non-constant meromorphic in C function is called elliptic, if it has two periods ul and linearly independent over R and generating the
lattice Q of all its periods. Without loss of generality we assume that
u2
Im— > 0.
The period parallelogram is a parallelogram with vertex at some point generated by vectors u and u2
In every period parallelogram a non-constant elliptic function takes every value the same number of times (the Liouville's theorem); the number is called the order of the elliptic function.
Let al,... ,an be zeroes of a non-constant elliptic function and bl,... ,bn be its poles lying in a period parallelogram (we take every zero or pole their multiplicity times). Then
al + ... + a„ = bl + ... + b,n (mod Q).
Further, we will denote by u an arbitrary element of the lattice Q.
One of the main elliptic functions is the Weierstrass P-function
1 NT^'
P(z ) = "2 + E
11
(z - u)2 u2
£ + g2 2 + g3 4 +
z2 + 20 z +28 z +
where g2 and are so-called Weierstrass invariants, defined by the equalities
60 = ^ u4, 140 = ^ u6;
the summation ' is over all non-zero elements of Q. The Weierstrass Z-function
z (z) = i + E'
+ " + —
= 1 _ 92 z3 _ z5 +
z 60z 140z +
z - u u u2
has the properties Z'(z) = -P(z),
Z(z + uk) = Z(z) + nk, k =1, 2, (5)
where nk = 2Z (uk/2). In every period parallelogram the function has a unique pole with residue 1.
The following relation is valid
u2nl - ulnl = 2ni. (6)
1
1
At last, we need the Weierstrass a-function
(z) = z n'{ (1 - z ) ex^ z + ê)] = z - 2420 z5 - 8S0 z7 +...
J )
a(z) = z^ 11 - Û) expl- + r = z - ^ ^ - 840 It is an odd entire function satisfying a'(z )
a(z)
= Z (z), a(z + —) = ea(z)en(z+w/2),
where n = m^i + nn2, if — = m—1 + n—2; e =1 if —/2 is a period and e = —1 otherwise.
Let us write the expansion of a function, connected with the Weierstrass Z-elliptic functions, at the origin:
<(-):= Z(z) - 1 = -60z3 - JLz5 + ... (7)
We will also need the following theorem by Weierstrass. It allows us to recover elliptic function by its zeroes and poles up to a multiplicative constant.
Theorem 3. Every elliptic function f can be represented in the form
,a(z - a1)... a(z - ar)
f (z ) = C
a(z - b1)... a(z - br)
where C = const, ai,..., ar is a full system of its zeroes, and bi,..., br is a full system of its poles, satisfying a1 + ... + ar = b1 + ... + br.
The following theorem gives explicit expressions of partial derivatives of the function log a(z; 0^,0^) with respect to 0 and o2.
Theorem 4. [13] We have
d log a(z; —1,—2) 1 d—1 2ni
d log a(z; —1,—2) 1
d—2 2ni
2 — 2(P(z) - (Z(z))2) + n2(zZ(z) - 1) + —2g z2 2 —1(p(z) - (Z(z))2) + m(zZ(z) -1) - —1 g z2
4. The differential equation for a one-parametric family of elliptic functions. Consider a one-parametric family f(z,t) of order n ^ 2 with periods o1(t) and o2(t), depending smoothly on a real
parameter t. Assume that f (z, t) has a unique (up to the period lattice) pole at z = 0 for every fixed t. Then f'(z,t) is an elliptic function with the same periods ul(t) and u2(t). According to Theorem 3, we have
f '(z,t)
N
n
j=0
(z - aj)
an+l(z)
where c = c(t) = 0, aj = aj(t) are pairwise non-equivalent (with respect to the period lattice Q) points, and the multiplicities mj satisfy the equality
Ej=o mj = n + 1
Differentiating by z and t the periodicity conditions
f (z + uk (t),t) = f (z,t), k =1, 2,
we obtain (in more detail, see [13]) Theorem 5. The function
hlz t) •= iM
satisfies
h(z + uk(t),t) - h(z, t) = —uk(t), k = 1, 2. (9)
Now we will recover the meromorphic function h(z, t) by its singularities for a fixed t. For the sake of simplicity, we assume that ul = 1; this does not restrict generality because the periods uk are defined up to a complex non-zero multiplier.
Theorem 6. Under the assumptions above, the family of functions f (z,t) satisfies the equation
f(z,t) _ 1 E Ak -lG?k(e) [Z(£) - Z(£ - z) - nlz]
f'(z, t) c (mk - 1)! -l
where ak = ak(t), Ak = Ak(t), nl = nl(t) is defined by (5), and
S=afc
Gk(z):= Gk(-M
Proof. Let us write the Taylor expansion of f (z,t) at ak:
f (z,t) = Afc + Bfc(z - afc)mk+1 + .
where
We have
where
B
k =
f(mk+1)(ak ,t) (mk + 1)!
f' (z,t)^7^ ((z - ak )mk + ...)
Gk (z)
Gk (z)
an+1(z)
therefore,
Bk
N
n (z - aj)
j=0,j=k
(mk + 1)Gk (ak)
From (11) it follows that
f '(z,t) = (mk + 1)Bk (z - ak )mk + ..., f(z,t) = Ak - (mk + 1)akBk(z - ak)mk +
and this implies
f(z,t) = Ak - (mk + 1)akBk(z - ak)mk + f '(z,t) c((z - ak )mk + ...)
Ak Pk,mk -1 (z) c (z - ak)
:ii)
:i2)
;i3)
;i4) :i5)
mk
Gk (z)
+ O(1), z ^ ak(t), (i6)
where Pk,mk-1(z) is the Taylor polynomial for Gk (z) of degree (mk - 1) at the point ak.
From (i6) we have
- Ak T-1 Gk')(ak)
*(--.«) = £ A £
c ^ s!(z - ak)mk-
k=1 s=0 V ky
+ O(1), z ^ ak. (i7)
Now we rewrite (i7) in the form
c
N A mk-1
h(z,t) = E A E (-!)mfc-S-1
k=1 s=0
X
x G^)((z - gfc)-1)(mk-S-1) + 0(1) X s!(mfc - s - 1)! + O(1), Z ^ ak• (18)
The function f(z,t) has a pole of order at most n at the point z = 0, and the order of the pole f'(z,t) is equal to (n + 1). Therefore, h(z,t) vanishes at z = 0. The function
1 N mfc-1
G(z,t) := h(z,t) - ^ E Ak E (-1)™'-S-1 X
k=1 s=0
x Gks)(ak)Z(mk-s-1,(z - ak) s!(mk - s - 1)!
has only removable singularities at all points of the lattice Q, and it is holomorphic at others points of the complex plane. Consequently, it can be extended, as a holomorphic function, to the whole plane C.
By (5) and (9), taking into account doubly-periodicity of the Weier-strass Z-function, we have
G(z + ^k, t) - G(z, t) = -¿k - E Ak G-^ • (19)
c k=i (mk -1)!
From (19) we see that at infinity G grows not faster than a linear function; therefore,
G(z,t) = az + p.
Thus,
1 N mfc-1
h(z, t) = 1 E Ak E (-1)™'-S-1 x
= S=0 X ^rrgrak'+az+p. (20)
From h(0, t) = 0 we find
o 1 V a V( 1)mk-s-1 Gks)(ak)Z(mk-s-1)(-ak) 0 = -1 Ik k5 (-1) - s - 1)! =
= 1 f A Tp Gks)(ak)z(mk-s-1)(ak) = c k=1 ks=0 s!(mk - s - 1)!
1 ^ ; (Gk(z)Z(z))(mk-1) Ak
c (mk - 1)!
(2i)
z=«k
From (i9) it follows that
nk a Gj (aj) 7 1 o roo^
a°k = -°°k-4- (m,-1)! , k = 1,2. (22)
Since o1 = o1(t) = 1, we obtain from (22)
a = -" £ ^k ^^ . (23)
c (mk - 1)!
Finally, from (20), (2i), and (23) we deduce (i0). □
5. System of equations for the parameters. Now we will find a system of ODEs for the parameters ak = ak (t), o2 = o2(t), and c = c(t).
Theorem 7. Let f (z,t) be a smooth one-parametric family of elliptic functions with periods o1 = o1(t) = 1 and o2 = o2(t) and let their derivatives have the form (8). Then the critical points aj = aj (t) of f (z,t), the period o2(t), and the multiplier c(t) in (8) satisfy
1
a k = -
c
. , G?kmk)(ak) dmk-1Gk(e)[Z(0 - C(e - ak) - mak] Ak
mk! (mk - 1)! d£mk-1
d mj -1
E t^ttw ^j-r g (e)[Z(e) - z(e - ak) - mak] j-
j
j=1, j=k
2ni A a Gkmk-1)(ak)
(24)
?=«j
02 = -X> , <25)
N
j=0
c = — c^^TOj Z(aj)äj + w2-
. d log a(aj)
+
N
+"E
Afc -1Gfc (e )(p(e) + m)
= (mfc — 1)!
de
mk -1
(26)
?=«k
the partial derivative d log a/d^2 in (26) can be found with the help of Theorem 4.
Proof. From (22) and (23) we find
N
^2
fc=1
^2^1 — n2 V^ j GG£mk 1)(afc)
(mfc — 1)!
and, with the help of the equality (6), we obtain (25). From (8), (15), and (10) we deduce
(mk + 1)akBk(z - ak)mk + ••• = Ak - h(z,t)f'(z, t) =
1
(z — afc)mk ömk-1Gfc(e) [z(e) — z(e — z) — ^ ]
Gfc (z)(mfc — 1)! demk-1
?=«k
N
namj (z — aj) N
"E Aj
j=o_^ Aj ömj-1Gj (e) [z (e) — z (e — z) — ^
an+1(z)
j=1, j=fc
(mj — 1)! dem
(27)
Now we will find the coefficient of (z — ak)mk in the last expression, which is the difference of two expressions, E1 and E2. For the second expression E2 we have
N
(z — a7-) N
/=0 ( j) £ -1 Gj (e)[z (e) — z (e — z) — mz]
an+1(z)
j=1, j=k v'"j y^ Aj w Aj d ~
(m,, — 1)! öemj-1
N
Aj (t) Aj -1Gj (e) [Z (e) — Z (e — afc) — mafc]
j=1, j=fc
X
(mj — 1)!
N
n <rmj (afc — aj)
j=o,j=fc
(mj — 1)! öemj-1
S=Oj
X
an+1(afc)
(z — afc)mk + o((z — afc)mk), z ^ afc.
c
Now we need an auxiliary result from [i2].
Lemma 1. Let f be n times differentiable at a point a, f (a) = 0 and Pj be the Taylor polynomial for f of degree j at a. Then
P„(x)=(x-f+im №)
n! d4n \x - 4/
Pn-1(x)
f (x)
1---
1 f (n)(a)
n! f(a)
(x - a)n + o((x - a)n), x ^ a.
For the first expression E1 we have
1 (z ak)- d— (4)[Z(4) - Z(4 - z) - n1z]
Gfc(z)(mfc - 1)! d4mk-1 k(4)[Z(4) Z(4 ) n1 ]
?=«fc
1
Gk(z)(mk - 1)! de""-
Ç=«fc
( k) ^ (3k(4)[Z(4)-3(4 - ak)-mafc]
(3k(ak)(mk - 1)! d4mfc_
+ o((z - ak )mk ),
Ç=«fc
therefore, with the help of Lemma i, the term with (z - ak)mk can be written as
(z - ak)mk Gk (ak )
Gkmk)(ak) d-1(3k(4)[Z(4) - 3(4 - ak) - mak]
mk !
(mk - 1)! d4mk-1
as z ^ ak, where 3(z) = Z(z) - 1/z. Thus, from (27) we deduce (24).
Finally, we will find a differential equation for c. From (8) and (10) we obtain the following expansion at z = 0:
N
f(z,i) = (-1r]>]
A k
d mfc
1
k=1
(mk - 1)! d4mk-1
G3k (4 )[P(4) + m]
X
ç=«fc
N
1
(aj ) - + O
j=o z
Thus,
1
A d mk-1 ~
f(z,i) - (-i)"+1n E (mr-ijT dfm-r G (f) PK) + ni]
X
?=«fc
N
1
XII ^ K") , z ^ 0. (28)
j=o Z
On the other hand,
N
f '(z,i) = (-1)n+1^amk (afc) -L + of^ fc=0 ^
and, therefore,
N
f '(z,t) - (-1)n+in ^ (a*)x
fc=0 N
x < c + c ^^ mj Z (aj)äj + j=o
. ö log a(aj)
^ra+1
z->- 0. (29)
Now, from (28) and (29) we obtain (26). □
Acknowledgment. This work was supported by the Russian Foundation for Basic Research, grant 18-41-160003.
The author is grateful to the anonymous referee for careful reading of the paper and useful remarks.
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Received August 30, 2018.
In revised form, November 26, 2018.
Accepted November 29, 2018.
Published online December 12, 2018.
Kazan Federal University
35 Kremlevskaya str., Kazan 420008, Russia
E-mail: snasyrov@kpfu.ru
