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§2. Symplectic Pairings on Lattices and Division Points 235

In the case of a complex torus T = C/L this subgroup N T = [(1/N)L]/L is
isomorphic to L/NL.If ω 1 , ω 2 is a basis of L over Z, then N T is isomorphic to
(Zω 1 /N + Zω 2 /N)/L. Thus the group of N-division points on a complex torus T
2
has order N and is the direct sum of two cyclic groups of order N. The group N T
is also the kernel of the isogeny N : T → T ,and L λ is the lattice with

λ L/L = ker(λ : C/L → C/L ).

(1.5) Remark. If λ : T = C/L → T = C/L is an isogeny of degree N, then
ker(λ) ⊂ N T and ker(λ = λ L/L where L ⊂ λ L ⊂ (1/N)L,[ λ L : L] = N,and
l λ

[(1/N)L : λ L] = N. Further, λ : T → T factors by T = C/L → C/ λ L →
λ

C/L = T , where C/ λ L → C/L is an isomorphism. Thus up to isomorphism, each
∗
∗
isogeny of T = C/L of degree N is given by a lattice L ⊃ L with [L : L] = N
l
and has the form C/L → C/L . A cyclic isogeny is one with the kernel a cyclic
∗
group.
§2. Symplectic Pairings on Lattices and Division Points
Let L be a lattice in C, and choose a basis such that L = Zω 1 + Zω 2 satisfies
Im(ω 1 /ω 2 )> 0. The determinant gives rise to a symplectic pairing e or e L defined
e : L × L → Z

which is given by the formula

a b
e(aω 1 + bω 2 , cω 1 + dω 2 ) = ad − bc = det .
c d

A change of basis from ω 1 , ω 2 to ω , ω is given by a 2 by 2 matrix of determinant

1 2
1, and the pairing associated with the second basis is the same as the pairing defined
by the first basis.
(2.1) Remark. The function e : L × L → Z is uniquely determined by the require-
ments that:
(1) e is biadditive, or Z-bilinear,
(2) e(x, x) = 0 for all x in L,

(2) e(x, y) =−e(y, x) for all x, y in L,and
(3) for some basis ω 1 , ω 2 of L with Im(ω 1 /ω 2 )> 0wehave e(ω 1 ,ω 2 ) = 1. This
holds then for any oriented basis of L.
Note that under (1) the two assretions (2) and (2) are equivalent. The assertion (3)

will hold for any such basis when it holds for one, and by (1), (2), (2) the above
formula for e in terms of the determinant is valid and shows the uniqueness.