# Elliptic functions proof - finitely many zeros and poles

• binbagsss
In summary, elliptic functions are complex-valued functions that are periodic in two directions and have many applications in mathematics and physics. They have a finite number of points where they are equal to zero or undefined, which is proven using complex analysis techniques. The finiteness of zeros and poles in elliptic functions has important applications in number theory, algebraic geometry, and physics, and is also used in the study of elliptic curves and cryptography. Elliptic functions cannot have an infinite number of zeros or poles due to their periodic and bounded nature.
binbagsss

## Homework Statement

Hi

I have questions on the attached lemma and proof.

##f(z)## is an elliptic function here, and non-consant ##\Omega## is a period lattice.
So the idea behind the proof is this is a contradiction because the function was assumed to be non-constant but by the theorem that if f is analytic in a region ##R## with zeros at a sequence of points ##a_i## that tend to ##a_0## ##\in R##, then ##f## is identically zero in ##R##.

Questions - mainly I don't understand where the consruction of the sequence comes from

- which of the conditions out of the three: bounded, closed, infinitely many zeros, means that a convergent sequence of zeros can be constructed? I don't understand the reasoning behind the sequence, and does it make use of the fact of the periodicity of ##f(z)##?
- I'm guessing this sequence would not be possible to construct if there were only finitely many zeros for the proof to work...?
- When it argues that by continuity ##f(a_0)=0##, we have that ##\Omega## has inifnitely many zeros as our assumption, but we haven't said anything about poles? We haven't disallowed poles, and this means the region is not analytic so not continuous and so the limit may not necessary exist?

see above

## The Attempt at a Solution

see above

#### Attachments

• lemm proof.png
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Hello,
Thank you for your post. In order to better understand the proof and the construction of the sequence, it would be helpful to have more context and information about the specific lemma and theorem being referenced. Can you provide more details or a link to the attached lemma and proof? This will allow me to give a more thorough and accurate response.

In general, the construction of the sequence of zeros in the proof likely relies on the periodicity of the function and the fact that it is non-constant. This allows for the creation of a sequence that converges to a point in the region, which then implies that the function is identically zero in that region.

Additionally, the conditions of boundedness, closedness, and infinitely many zeros are all important in the construction of the sequence. Without all three of these conditions, the proof may not hold. If there were only finitely many zeros, the proof may not work because the sequence would not necessarily converge to a point in the region. If the region was not closed, there may be points outside of the region where the function is not zero, which would contradict the assumption that the function is identically zero in the region.

I hope this helps. Please provide more information about the specific lemma and proof so that I can give a more detailed response. Thank you.

## 1. What are elliptic functions?

Elliptic functions are complex-valued functions that are periodic in two directions. They are closely related to elliptic curves and have many applications in mathematics and physics.

## 2. What does it mean for an elliptic function to have finitely many zeros and poles?

It means that the function has a finite number of points where it is equal to zero or where it is undefined. In other words, the function has a finite number of points where its value becomes infinite.

## 3. How is the proof for the finiteness of zeros and poles in elliptic functions performed?

The proof for the finiteness of zeros and poles in elliptic functions is usually done using complex analysis techniques, such as the argument principle and the maximum modulus principle.

## 4. What are the applications of the finiteness of zeros and poles in elliptic functions?

The finiteness of zeros and poles in elliptic functions has important applications in number theory, algebraic geometry, and physics. It is also used in the study of elliptic curves and their applications in cryptography.

## 5. Can elliptic functions have an infinite number of zeros or poles?

No, elliptic functions cannot have an infinite number of zeros or poles. This is because they are periodic and bounded, which means that they can only have a finite number of zeros and poles in any given region of the complex plane.

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