Let w1,w2,w3 be independent periods for a meromorphic function f. This means that if
i=13aiwi = 0 for each ai an integer, then each ai = 0. Hint: At some point you may want to use
- Show that if ai is an integer, then ∑
i=13aiwi is also a period of f.
- Let PN be periods of the form ∑
i=13aiwi for ai an integer with
≤ N. Show there
3 such periods.
- Show PN ⊆
2 ≡ QN.
- Between the cubes of any two successive positive integers, there is the square of a positive
3 < M2 <
3. Show this is so. It is easy to verify if you show
3∕2 − x3∕2 > 2 for all x ≥ 2 showing that there is an integer m between
3∕2 and n3∕2. Then squaring things, you get the result.
- Partition QN into M2 small squares. If Q is one of these, show its sides are no longer
- You have
3 points which are contained in M2 squares where M2 is smaller than
2. Explain why one of these squares must contain two different periods of
- Suppose the two periods are ∑
i=13aiwi and ∑
i=13âiwi, both in Q which has sides of length no
more than C∕N1∕2. Thus the distance between these two periods has length no more than
C∕. Explain why this shows that there is a sequence of periods of
f which converges to 0.
Explain why this requires f to be a constant.
This result, that there are at most two independent periods is due to Jacobi from around 1835. In
fact, there are nonconstant functions which have two independent periods but they can’t be