Let C be the algebraic curve defined by the modular polynomial &#x03D5;<!-- ϕ -->

Kenley Wagner

Kenley Wagner

Answered question

2022-05-21

Let C be the algebraic curve defined by the modular polynomial ϕ N of order N > 1 over the rational numbers, i.e.
C := specm ( Q [ X , Y ] / ϕ N ( X , Y ) ) .
The singularities of this curve can be removed and we obtain a nonsingular curve C s n n, then, we can embed C s n into a complete non-singular curve C ¯ .
In Milne's notes "Modular Functions and Modular Forms" it is written:
The coordinate functions x and y are rational functions on C ¯ , they generate the field of rational functions on C ¯ , and they satisfy the relation ϕ N ( x , y ) = 0.
I assume, by coordinate functions he means the functions f ( X ) , g ( X ) Q [ X ] such that ϕ N ( f ( x ) , g ( x ) ) = 0 for all x Q. However, I don't understand why the field of rational functions on C ¯ is generated by these functions. Could someone explain this to me?
Thank you very much in advance!

Answer & Explanation

zepplinkid7yk

zepplinkid7yk

Beginner2022-05-22Added 11 answers

The coodinate functions x and y are the image of X and Y in the quotient of Q [ X , Y ]. The map ξ : Q [ X ] Q [ X , Y ] / ( ϕ N ( X , Y ) ) obtained from the composition Q [ X ] Q [ X , Y ] Q [ X , Y ] / ( ϕ n ( X , Y ) ) will correspond to the map x : C S p e c Q [ X ] = A Q 1 , which is a regular function on the non-singular locus C s n of the completed curve C ¯ , in other words a rational function.
That x and y satisfy the relation ϕ n ( x , y ) is just by definition.
The injection Q [ X ] Q [ X , Y ] corresponds to the projection A Q 2 A Q 1 onto the x coordinate, assuming we identify the points of A Q 2 with pairs ( x , y )). If C is a curve living in A Q 2 , the map above is just its restriction to C, so it's the map giving the x coordinate of the points on C.

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