- Show the set of modular transformations is a group. Also show those modular
transformations which are congruent mod2 to the identity as described above
is a subgroup.
- Suppose f is an elliptic function with period module M. If and
are two bases, show that the resulting period parallelograms
resulting from the two bases have the same area.
- Given a module of periods with basis and letting a typical element of this
module be denoted by
w as described above, consider the product
Show this product converges uniformly on compact sets, is an entire function, and
where ζ was defined above as a primitive of
℘ and is given by
- Show ζ =
ηi where ηi is a constant.
- Let Pa be the parallelogram shown in the following picture.
dz = 1 where the contour is taken once around the
parallelogram in the counter clockwise direction. Next evaluate this contour integral
directly to obtain Legendre’s relation,
- For σ defined in Problem 3, 4 explain the following steps. For j = 1,2
Therefore, there exists a constant, Cj such that
Next show σ is an odd function, and then let
z = −wj∕2 to find
Cj = −e
- Show any even elliptic function, f with periods w1 and w2 for which 0 is neither a
pole nor a zero can be expressed in the form
where C is some constant. Here ℘ is the Weierstrass function which comes from the
two periods, w1 and w2. Hint: You might consider the above function in terms of
the poles and zeros on a period parallelogram and recall that an entire function
which is elliptic is a constant.
- Suppose f is any elliptic function with a basis for the module of periods.
57.1.8 and 57.3.41 show that there exists constants a1,
,bn such that for some constant C,
Hint: You might try something like this: By Theorem 57.1.8, it follows
that if are the zeros and
the poles in an appropriate period
bk equals a period. Replace αk with ak such that
bk = 0. Then use 57.3.41 to show that the given formula for f is bi
periodic. Anyway, you try to arrange things such that the given formula
has the same poles as f. Remember an entire elliptic function equals a
- Show that the map τ → 1 − maps
l2 onto the curve, C in the above picture on
the mapping properties of λ.
- Modify the proof of Theorem 57.1.22 to show that λ