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162 Chapter 6. Complex multiplication in higher generaA theta function of the form z ↦→ e Q(z) where Q is a polynomial of degree at most two is saidto be trivial and two theta functions are said to be equivalent if their quotient is a trivial thetafunction.The applicationΛ × V → C(λ, z) ↦→ a λ (z)can be uniquely extended to an application a : V × V → R which is R-linear in the first variableand C-linear in the second one. The R-bilinear alternating formis called the Riemann form associated with θ.ω(x, y) = a(x, y) − a(y, x)Proposition 6.1.6 ([65, Propositions IV.1.3, IV.1.5]). With the above notation, the Riemannform ω is real valued, integer valued on Λ and satisfiesω(ix, iy) = ω(x, y)for all x and y in V . Moreover ω(x, ix) ≥ 0 for all x ∈ V ; we say that ω is positive.If ω(x, ix) > 0 for all x ∈ V different from O, then we say that ω is positive definite ornon-degenerate.A Hermitian form, or symmetric sesquilinear form, H is associated with the Riemann form ω:H(x, y) = ω(x, iy) + iω(x, y) .It is C-antilinear in the first variable and C-linear in the second one 5 . Then ω is nothing butω = I(H) and there is a one-to-one correspondence between this two kinds of forms.It is easily seen that any theta function can be normalized, i.e. multiplied by a trivial thetafunction, to satisfyθ(z + λ) = α(λ)e πH(λ,z)+ π 2 H(λ,λ) θ(z)where α : Λ → C 1 = {z ∈ C | |z| = 1} is a semicharacter:α(λ 1 + λ 2 ) = α(λ 1 )α(λ 2 )(−1) ω(λ1,λ2) ,called the multiplicator, or canonical factor of automorphy, of θ. The theta function θ is said tobe of type (H, α).It can be shown that any complex embedding of a complex torusu : V/Λ → P ncan be described by n + 1 theta functions of the same type u = (θ 0 , . . . , θ n ). Hence there exists aRiemann form on Λ whose kernel must be trivial, i.e. the form is positive definite, or equivalentlynon-degenerate.A theta function θ on the complex torus X = V/Λ, or the associated hermitian form H andmultiplicator α, can be used to construct a line bundle L(H, α) as the quotient of V × C by theaction of Λ given byλ · (z, t) = (z + λ, α(λ)e πH(λ,z)+ π 2 H(λ,λ) t) .The Appel–Humbert theorem states that the converse is true: every line bundle can be constructedin this way.5 This is the convention of Debarre [65, III.1.1]. Mumford [213, I.2], Lang [159, 1], Birkenhake and Lange [16,Lemma 2.1.7], and Hindry and Silverman [132, A.5] swap x and y, i.e. set ω(ix, x) positive and H(x, y) =ω(ix, y) + iω(x, y). Shimura [238, I.3.1] sets ω(x, ix) positive and 2iH(x, y) = ω(x, iy) + iω(x, y) which is skewhermitian.Milne [205, I.2.3] sets ω(x, ix) positive and H(x, y) = ω(x, iy) − iω(x, y).