The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point (0,1,0)) still seems unclear.For example, Silverman (in The Arithmetic of Elliptic Curves) states that the order of the rational function y on the elliptic curve y 2 = ( x − e 1 ) ( x − e 2 ) ( x − e 3 ) ,where e 1 , e 2 , and e 3 are distinct, is −3. That is, the function y has a pole of order 3 at (0,1,0). I have no doubt that this is true; I'd like to know a simple way to see it, based on projective coordinates and independent of the fact that the sum of the orders of the zeros of y is 3 (which I understand).

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The matter of explicitly finding the order of a rational function on an elliptic curve in the projective plane at infinity (i.e. at the point (0,1,0)) still seems unclear.For example, Silverman (in The Arithmetic of Elliptic Curves) states that the order of the rational function   y on the elliptic curve      y    2    =  (  x  −      e    1    )  (  x  −      e    2    )  (  x  −      e    3    )  ,where       e    1  ,       e    2  , and       e    3   are distinct, is −3. That is, the function   y has a pole of order 3 at (0,1,0). I have no doubt that this is true; I&#039;d like to know a simple way to see it, based on projective coordinates and independent of the fact that the sum of the orders of the zeros of   y is 3 (which I understand).

Kaylyn Ewing · Accepted Answer

This fairly old-fashioned argument is the way I look at the situation. Homogenize       y    2    =      x    3    +  a  x  +  b to       Y    2    Z  =      X    3    +  a  X      Z    2    +      Z    3  , then dehomogenize by setting   Y  =  1 to get   ζ  =      ξ    3    +  a  ξ      ζ    2    +      ζ    3  . The point you’re interested in is now (0,0).What does the curve look like there? Clearly   ξ is a local uniformizer, and   ζ has a triple zero there. If you don’t see that right away, use the equation relating the two letters to see that   ζ expands as a power series that starts   ζ  =      ξ    3    +  a      ξ    7    +  ⋯. Now, what are the original   x and   y in terms of   ξ and   ζ? Yes:   x  =  ξ      /    ζ and   y  =  1      /    ζ, and there you are.

The matter of explicitly finding the order of a rational function on an elliptic curve in the projec

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