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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?

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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).

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