Valence formula

Stream: maths

[Chris Birkbeck]

How far are we from being able to prove the valence formula for level 1 weakly meromorphic forms? (See for example Theorem 1.6.6 on page 16 of this: https://mdave16.github.io/notes/Modular%20Forms%20-%20Marc%20Masdeu.pdf). The proof requires one to, amongst other things, compute a contour integral, where the contour is defined by lines and circle segments. I know we can do squares and circle contours but can we do more general ones?

The reason I ask is that from this one can prove that the spaces of level one modular forms are finite dimensional. Moreover, as in the attached notes (as I was recently reminded) one can prove that the dimensions are finite for other levels as well by using this. Importantly this wouldn't require Riemann--Roch or GAGA to do this.

[Yury G. Kudryashov]

The main reason why I used circles and rectangles is that it's not that trivial how to say that a point is inside the region bounded by some curve.

[Chris Birkbeck]

Aha good point. I can definitely see this being annoying. How good of a description do you think we need for this? For example, in this case, one can (I think) describe the contour as a rectangle with some open balls removed. So one could say a point is in my contour iff its in this rectangle but not in these balls. Do you think such a thing would be too annoying to work with?