What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
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1$\begingroup$ Obviously infinite-sheeted branched covers only exist in the noncompact setting. For examples, just take the (unbranched) universal cover of a compact surface of positive genus. $\endgroup$– Andy PutmanCommented Jun 26 at 10:09
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$\begingroup$ Thanks But I am looking for a branched (not unbranched) cover ;) $\endgroup$– cataCommented Jun 26 at 10:29
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2$\begingroup$ Then lift a branched cover between compact Riemann surfaces to their universal covers. $\endgroup$– Andy PutmanCommented Jun 26 at 10:30
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2$\begingroup$ For instance, the $j$-function mapping the upper half-plane onto the complex plane. $\endgroup$– Moishe KohanCommented Jun 26 at 10:51
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1 Answer
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For infinite degree, the definition of "branched covering" can be somewhat ambiguous. But $$z\mapsto \cos z: \mathbb{C}\to \mathbb{C}$$ $$\wp: \mathbb{C}\to S$$ are a simple examples of branched coverings, according to any definition. Here $S$ is the Riemann sphere. They are regular (factors by some groups). Here is an example of non-regular one: $$z\mapsto ze^z: \mathbb{C}\to \mathbb{C}.$$
answered Jun 26 at 14:00