Math 621 Homework Assignment 3 Spring 2000

Due: Tuesday, March 7

Problems 3, 7, 8, 10, 12 are recommended, but you need not hand them in.

1. 1. Find a function H(x,y) harmonic in the domain and such that on the line and on the line y=1.
   2. Find a function G(x,y) harmonic in the region inside the circle and outside the circle and such that on the inner circle and on the outer circle. Hint: See HW 2 problem 2(a).
2. Lang page 238-239 problems 12 part c (interpret the result geometrically), 14 parts b,c, 13 part c (classify the fixed points according to Problem 14).
3. Let C and C' be circles in the complex plane. Let and be the corresponding reflections (sending a point z to the symmetric point ). Show that if T is a fractional linear transformation, then is a fractional linear transformation.
4. Ahlfors page 83 problem 2: Reflect the imaginary axis, the line x=y, and the circle in the circle .
5. Ahlfors page 83 problem 4: Find the linear fractional transformation which carries the circle into , the point -2 into the origin, and the origin into i.
6. Ahlfors page 83 problem 5: Find the most general linear fractional transformation of the circle into itself.
7. Ahlfors page 83 problem 6: Suppose that a linear transformation carries a pair of concentric circles into another pair of concentric circles. Prove that the ratio of the radii must be the same.
8. Ahlfors page 83 problem 7: Find a l.f.t which carries and into concentric circles. What is the ratio of the radii?
9. Ahlfors page 33 problem 4: What is the general form of a rational function (of arbitrary degree) which has absolute value 1 on the circle ? In particular, how are the zeros and poles related to each other? (You may use the Fundamental Theorem of Algebra stated in Corollary 7.6 page 130 in Lang. We will prove it later in the course). Hint: Prove first that the rational function f satisfies , where R is the reflection with respect to the unit circle.
10. Ahlfors page 33 problem 5: If a rational function is real on , how are the zeros and poles situated?
11.   Let be the permutation group of the set . For each permutation , denote by the fractional linear transformation taking to .
    1. Find the six l.f.t .
    2. The orbit of is the set . Show that all but two orbits in consist of six elements. One special orbit in consists of two elements and the other special orbit consists of three elements.
12.   We have seen that the cross ratios and are equal, if and only if there exists a linear fractional transformation mapping the first ordered 4-tuple to the second. We work out the analogous statement for unordered sets of 4 distinct points. Let j be the rational function (of degree 6)

    

    The composition , of j and the cross ratio, is called the j-invariant of the unordered set . Use your answer to problem 11 to show

    1. The function is symmetric in the .
    2. There exists a linear fractional transformation mapping the unordered set onto if and only if their j-invariants are equal.
13. Lang Ch III Sec 2 page 102 problems 5, 7
14. 1. Describe the curve C parametrized by , . Compute .
    2. Compute .
15. 1. Let denote the semi-circle

       

       Show that

    2. Let be such that and . Show that

       

The following three challenge problems are a continuation of problem 12. They will be due only if and when we study elliptic functions. You have the knowledge required to answer the questions, so if you can afford the time, enjoy solving them.

1.   Given a complex number , we define the elliptic curve to be the cubic ``curve'' in given by the equation

    

   Let f be a l.f.t mapping the set to itself (see problem 11). Show that there is a constant c, depending of f and uniquely up to sign, for which the map

    

   is invertible and it maps the corresponding open set of onto that of . (Show it for two l.f.t generating and explain why it follows for all six).

   Note: is a surface in . A basic result in the theory of elliptic functions shows to be homeomorphic to the surface of a donut with one point removed (Theorem 2.3 page 399 in Lang). The complete elliptic curve is defined in a way analogous to the extended complex plane. Simply, regard and as two open sets of and identify with via (2) (with f(z)=1/z). Problems 11, 12 and 1 imply that the complete elliptic curve depends only on the j-invariant of .

2. Next, define the notion of an (oriented) angle between two intersecting arcs on as follows.
   1. Let , i=1,2, be two differentiable maps lying on and suppose that they intersect at the point p, . Show that their tangent vectors at p are linearly dependent as complex vectors in .
   2. If and , we define the oriented angle from to to be .
   Show that (2) is a bijective conformal map with respect to the above definition of angles. Show also that the map given by is conformal away from .

   Note: Two elliptic curves with distinct j-invariants can not be related by a bijective (orientation preserving) conformal map.

3. 1. Show that the set of invertible orientation preserving conformal automorphisms of an elliptic curve is a group and the subset fixing a point (say at infinity) is a subgroup.
   2.   Let G be the group of orientation preserving conformal automorphisms of an elliptic curve fixing the point . Show that G has at least two elements. If , G has at least 4 elements. If , G has at least 6 elements. Hint: See problem 11.

   Note: It can be shown that the number of elements in G is precisely the one indicated in Part 3b.

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