What is the relationship between holomorphic maps and elliptic functions?

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In summary, the conversation discusses finding a holomorphic map from C/(Z+iZ) to C/(Z+iZ) and whether it needs to be doubly periodic or an elliptic function. It is established that the function must map Z+iZ to itself and can be a simple function like the identity or a more interesting one like an elliptic function. The conversation also touches on the concepts of group homomorphisms and automorphisms of the torus. Finally, it is confirmed that the map must be complex linear and therefore the function is constant.
  • #1
Ant farm
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grrr, so annoyed, can't see the wood from the trees on this problem!
I'm trying to get a holomorphic map from C/(Z+iZ) -> C/(Z+iZ) where C=complex numbers and Z=integers.
Does this function have to be doubly periodic?
Are doubly periodic functions the same as elliptic functions?
Are all elliptic functions meromorpic but not holomorphic in which case I'm obviously not looking for an elliptic function!
Please Help
 
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  • #2
what is wrong with the identity function?
 
  • #3
It Can't be the Identity, must be something else with f(0)=0
I've been thinking of mayb a piecewise function involving the Weirestrass P function... or a rotation... I just can't see where I'm working in my head!
 
  • #4
What's wrong with z-->2z? It just needs to be some function that maps Z+iZ into itself.
 
  • #5
Ah, ok, sorry, I was looking for something that was doubly periodic, but since the function is going form Z+iZ into itself, that condition will automatically be satisfied?! And that is Holomorphic too.
It just seems too simple!
 
  • #6
If you want something 'interesting' then you will need elliptic functions, or doubly periodic ones, for sure - but there are simple ones too.
 
  • #7
i think these are actually all group homomorphisms, so think in those terms, i.e. any group map C-->C that takes Z+iZ to itself.

i fact the map C-->C is complex linear I believe.

An elliptic function is a holomorphic map from C/Z+iZ-->P where P is the projective line C U {pt}.

they form a field. the holomorphic maps from the torus to itself form a ring, and of course the units in that ring, the automorphisms of the torus, form a group. this group may be always as simple as Z/2Z/, Z/4Z, or z/6Z.
 
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  • #8
a holomorphic map f:C/Z+iZ-->C/Z+iZ, defines a holomorphic map g(z) =
f(z)-f(z+i) with values in the discrete set Z+iZ, hence g is constant.

then g' is zero, so f is periodic with period i, and simiklarly 1.

then f' is bounded hence constant, so f is linear and since f(0) = 0, f(z) = ax for some a in C.
 

1. What is a holomorphic map?

A holomorphic map is a function that is complex differentiable at every point in its domain. This means that it has a derivative at every point, which is also a complex number. In other words, it is a function that can be continuously and smoothly extended to the complex plane.

2. What is the significance of a holomorphic map?

Holomorphic maps are important in complex analysis, as they preserve the geometric properties of the complex plane. They also have many real-world applications in physics, engineering, and other fields.

3. How is a holomorphic map different from a regular function?

A holomorphic map is different from a regular function in that it is defined on the complex plane and has complex derivatives. Regular functions, on the other hand, may only be defined on the real numbers and have real derivatives.

4. What is the relationship between holomorphic maps and analytic functions?

Holomorphic maps are a special type of analytic function, which means that they can be expressed as a convergent power series. In fact, holomorphic maps are the only type of analytic function that is also complex differentiable at every point.

5. How are holomorphic maps used in complex geometry?

Holomorphic maps are used in complex geometry to study the geometric properties of complex manifolds. They are used to define holomorphic vector bundles, which are important objects in the field of algebraic geometry.

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