By H. P. F. Swinnerton-Dyer

The examine of abelian manifolds kinds a average generalization of the speculation of elliptic services, that's, of doubly periodic capabilities of 1 complicated variable. while an abelian manifold is embedded in a projective area it truly is termed an abelian type in an algebraic geometrical feel. This advent presupposes little greater than a uncomplicated path in complicated variables. The notes include the entire fabric on abelian manifolds wanted for program to geometry and quantity thought, even if they don't include an exposition of both program. a few geometrical effects are incorporated in spite of the fact that.

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**Extra resources for Analytic Theory of Abelian Varieties **

**Example text**

This suggests adding k + 1 times the expression 26 1 Riemann Surfaces ( nr−1 ) φPi ∏t=0 Pt (which is 0 and does not change the value), but the divisor thus obtained is of degree g + n − 1 + (k + 1)rn and contains some points (P0 , the points in A, and the points in every set C j with 0 ≤ j ≤ k − 1) to the power n or higher. Therefore we shall subtract φPi (Pln ) for every such point Pl which now appears to the power n or higher (which is again 0 and does not change the value), and obtain ( ) ( ) nr−1 k+1 nr−1 Pt ∏t=0 n−1 k+1 n−1 + KPi = φPi P0 ∆ n + KPi .

3 Theta Functions 17 [ ] (α + γ )/n ( )] θ (ζ , Π ) (α ′ + γ ′ )/n γt α′ − β ′ [ ] , e − 4n n (β + γ )/n θ ζ , Π ) ( (β ′ + γ ′ )/n [ which is a root of unity of order 4n2 . If we take the quotient to the power n2 , we get the previous expression equal to [ ] 2 (α + γ )/n n [ ] θ (ζ , Π ) (α ′ + γ ′ )/n γt [ ] e − (α ′ − β ′ ) . 4 (β + γ )/n θ n2 ζ , Π ) ( (β ′ + γ ′ )/n The final observation for now is that if γ and the difference α ′ − β ′ are in 2Zg , then the multiplier disappears, a fact which will be written as [ ) [ ]( ] γ′ γ 2 α /n (α + γ )/n n2 , θn ζ + Π Π θ (ζ , Π ) + I (α ′ + γ ′ )/n α ′ /n 2n 2n [ ) [ ]( ] = .

Now, applying Eq. 1) again to our divisor ∆ of degree g − 1 (integral or not, this does not matter) we can easily obtain φQ (∆ ) + KQ = (g − 1)φQ (P0 ) + φP0 (∆ ) + KQ = φP0 (∆ ) + KP0 , which completes the proof. 12 is the relation between the two vectors of Riemann constants KP0 and KQ which appears in the proof, regardless of the divisor ∆ . 12 will involve divisors ∆ of degree g − 1, and this is why its assertion is formulated in this way. 12 for non-integral divisors as well, and it is good to know that it holds equally there (and without any additional complication in the proof).