Analytic Number Theory by Diamond H. (ed.)

By Diamond H. (ed.)

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If the total degree of a polynomial P in n variables is r, then P ≺ size(P )(1 + x1 + · · · + xn )r . We also need some facts about derivations. Recall that a derivation D of a ring R is a map D : R → R such that D(x + y) = D(x) + D(y) and which satisfies D(xy) = D(x)y + xD(y). Sometimes, we write Dx for D(x) when the meaning is clear. For instance, if R is the polynomial ring K[x1 , . . , xn ], then the partial derivative ∂/∂xi is a derivation. If R is an integral domain and K its quotient field, then a derivation D of R can be extended in the usual way by setting D(u/v) = vD(u) − uD(v) .

Murty and P. 1007/978-1-4939-0832-5 7, © Springer Science+Business Media New York 2014 27 28 The Six Exponentials Theorem Remark. There is the four exponentials conjecture of Schneider that says the theorem should still be valid if y1 , y2 , y3 are replaced by y1 , y2 linearly independent over Q. We refer to the interested reader a paper of Diaz [44] where he investigates the interrelation between values of the modular j-function (which we shall be defining later) and the four-exponential conjecture.

Let g2 , g3 be defined as above. Then all the complex solutions of the equation y 2 = 4x3 − g2 x − g3 are given by (℘(z), ℘ (z)) where ℘ is the Weierstrass ℘-function attached to L and z ranges over all the complex numbers in C\L. 4 Let f be an elliptic function associated with the lattice L and let resw f denote the residue of f at z = w. If D is a fundamental domain of f whose boundary ∂D does not contain any pole of f , then resw f = 0. w∈D Further, if ordw denotes the order of f at z = w and ∂D does not contain a zero of f , then ordw f = 0.

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