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§3. Isogenies in the General Case 237

(2.4) Proposition. Let E and E be elliptic curves over C. The isomorphism N E(C)
= [(1/N)L]/L → L/N L, given by multiplication by N, transfers the symplectic
pairing on Lto a nondegenerate pairing

e N : N E(C) × N E(C) → Z/NZ.

For an isogeny λ : E → E we have

e N (λx, x ) = e N (x, ˆ λx )

for x in N E(C) and x in N E (C).

A final statement about the complex case for the reader with a background in
homology theory. On H 1 (E(C), Z) = L, where E(C) = C/L, the symplectic ho-
mology intersection pairing on the real surface E(C) is e : L × L → Z in (2.1).

§3. Isogenies in the General Case

(3.1) Remark. For a lattice L we denote by El(L) the field of elliptic functions on

C/L.If λ : C/L → C/L is an isogeny, then f (z) → f (λz) defines an embedding
of El(L ) into El(L) as a subfield. The group λ −1

L /L acts on the field El(L) by
translation of variables in the function. The fixed field is equal to El(L ), or, more

precisely, the image of El(L ) in El(L). Hence El(L)/El(L ) is a Galois extension of

fields with Galois group λ −1
L /L = ker(λ), and by the theorem of Artin

L /L) = [L : λL].
[El(L) :El(L )] = #(λ −1

In fact, the embedding El(L ) into El(L) determines λ : C/L → C/L , and this
gives us the clue as to how to formulate the notion of isogeny in the general case.
Further, note that although an isogeny C/L → C/L is additive, it suffices for it to

be just analytic preserving 0 by the remarks preceding (1.1).
(3.2) Definition. For an elliptic curve E defined over a field k by the Weierstrass
equation f (X, Y) = 0 the function field k(E) of E over k is the field of fractions of
the ring k[X, Y]/( f ).
The field k(E) can also be described as the quadratic extension k(x, y) of the field
of rational function in one variables k(x) where y satisfies the quadratic equation
f (x, y) = 0in y over k(x).Inthe case E(C) = C/L over the complex numers
we know there is an isomorphism between C(E) and El(L) by 9(3.3), for El(L) is

generated by ℘ and ℘ and ℘ satisfies a quadratic equation of Weierstrass type over
the rational function field C(℘).
Now we survey the theory of isogenies over an arbitrary field and suggest reader
go to Mumford, Abelian Varieties.