I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm par

Aidyn Cox

Aidyn Cox

Answered question

2022-05-19

I've never fully understood the connection between Riemann surfaces and algebraic varieties. I'm particularly interested in the case of the modular curve of level N--I know how the Riemann surface is constructed by taking a quotient of the upper half-plane by the action of a congruence subgroup of the modular group, but not how the resulting manifold translates into a curve. From what I've read, it appears that the associated curve is defined by equations satisfied by functions defined on the manifold, but I don't understand which functions are involved in these equations. What exactly is the relation between the two types of objects?

Answer & Explanation

Makai Blackwell

Makai Blackwell

Beginner2022-05-20Added 11 answers

To see that some given Riemann surface is an algebraic curve, you usually need Riemann-Roch. A prototypical example of its use is the usual proof of existence of a Weierstrass equation for an elliptic curves (see e.g. Silverman).
But for the modular curves, things are actually simpler. It is easy to see that a modular curve is a covering of Γ ( 1 ) H . Now, the latter is the simplest Riemann surface there is, the Riemann sphere, and that is clearly an algebraic curve. There are lots of ways of seeing that Γ ( 1 ) H is the Riemann sphere, some of which are sketched in Milne's notes. An explicit isomorphism of Γ ( 1 ) H with C is provided by the j-function.
You can now use the covering to obtain an equation for a modular curve. For the minute details of this calculation for the case Γ 0 ( N ), see Milne's notes, Theorem 6.1. This does not need Riemann-Roch, and only relies on explicit computations with the j-function.
A more powerful approach to modular curves, which will also give you more information about fields of definition, is by interpreting the modular curves are moduli varieties.

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