Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic curves it is possible to define the degree as the number of poles (with multiplicities). Why is this the same? Can you give a reference?

share|cite|improve this question

I don't think that these are exactly the same, rather, the second is an instance of the first. A meromorphic function on an elliptic curve $C$ defines a morphism $f:C\to\mathbb P^1,$ and the number of poles is by definition the number of points in the fibre over $\infty$ (with multiplicities). The degree $\deg(f)$ is in general the number of (distinct) points in a general fibre of $f,$ which equals the number of poles, as long as we count with multiplicity. So it stands to reason that the two definition are the same.

A reference is section II.2 of Silverman (specifically, Example 2.2 and Proposition 2.6 should be helpful).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.