# An extension to Picard's theorem?

Posted by Mathoman, Monday 15 June 2009 at 17:38 - Conjecture

At the end of my article *Hyperelliptic action integral*, Annales de l'institut Fourier 49(1), p.303â€“331, I state the following conjecture:

Conjecture about a singularity

Let D be the open unit disk in the complex plane and be a cover of the punctured disk D*= D\{0} by open connected sets. Suppose on each open set there is an injective holomorphic function such that on every intersection . Then those differentials glue together to a meromorphic 1-form on D.

It is evident that the 1-form is holomorphic on *D**. In the case that its residue at the origin vanishes, the conjecture follows because the singularity can be at most a pole, as can easily be derived from Picard's big theorem (cited below). But I am unable to prove the conjecture for the case of a non-zero residue.

I would be grateful for any proof or counter-example — well, trule spoken, a counter-example would be less welcome since I hope that the conjecture is true. I am guided by my simple geometric intuition on Riemann surfaces...

In 1880 Charles Emile Picard (1856-1941) proved the following theorem.

Picard's Big Theorem

A holomorphic function takes in any neighborhood of an essential singularity every complex value, with at most one exception, infinitely often.

**A typical example for Picard's theorem**

is holomorphic on and has an essential singularity in . Is there a value that is does not take (Picard says "at most one exception")? Yes, since for all , there is an exceptional value and it must be zero; according to Picard's theorem there are, for any complex value and any , infinitely many complex numbers such that and .

**Direct compuation of this example**

For let and Thus we have

Hence

By taking big enough we see that has infinitely many pre-images in the punctured disk

**A less simple example**

*P*the set of all prime numbers and consider the following function

Since there is an essential singularity we can apply Picard's Theorem.

But a direct computation seems impossible...