New PDF release: Algebraists' Homage: Papers in Ring Theory and Related

By S. A. Amitsur, D. J. Saltman, George B. Seligman

ISBN-10: 082185013X

ISBN-13: 9780821850138

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Extra resources for Algebraists' Homage: Papers in Ring Theory and Related Topics

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3 Elliptic functions 25 iv) Show that, given a point and its domain of definitions, a meromorphic function can not have the same value at all points of a set of points that converges to that point. v) Show that if a meromorphic function f : C → C has two periods ω1 and ω2 that are linearly dependent over the reals but independent over the rationals, then f is constant. This exercise shows that if the two periods ω1 and ω2 lie on the same line (through the origin) we either have a completely trivial case or one which really has only one period.

Paths in the plane intersect at a point different from the base point; as paths in X they do not intersect since they represent two different square roots at that point. 6 Note that we traverse the line between the points twice in different directions but also with different choices of square roots so that the full integral is twice the integral along the line. 5 We shall soon see why. 6 We noted before that we may change base point by moving back and forth along a fixed path between the two base points and that the integrals along the fixed path cancel.

26 3 Elliptic functions ω1 ω1 + ω2 ω2 Figure 17. The fundamental domain and its identifications. Proof. Let f be such a function. A holomorphic function is continuous and hence so is z → |f (z)|. As such it has a maximum on the closed and bounded set F . Let a be a point where the maximum is achieved. As all the values of f are attained already on F , we get that |f (z)| ≤ |f (a)| for all z. By definition we may write f as ∞ an (z − a)n f (z) = n=0 in a disc around a. If all an = 0 for n > 0, then f is constant in the disc and then it is constant everywhere by Exercise 11 iii).

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Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman

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