Elliptic Functions according to Eisenstein and Kronecker by Andre Weil

By Andre Weil

"As a contribution to the background of arithmetic, it is a version of its sort. whereas adhering to the fundamental outlook of Eisenstein and Kronecker, it offers new perception into their paintings within the gentle of next advancements, correct as much as the current day. As one could anticipate from this writer, it additionally comprises a few pertinent reviews having a look into the long run. it's not in spite of the fact that only a bankruptcy within the historical past of our topic, yet a wide-ranging survey of 1 of the main energetic branches of arithmetic this day. The booklet has its personal very person flavour, reflecting a type of mixed Eisenstein-Kronecker-Weil character. established primarily on Eisenstein's method of elliptic features through limitless sequence over lattices within the advanced airplane, it stretches again to the very beginnings at the one hand and reaches ahead to a few of the latest examine paintings at the different. (...) The continual reader should be richly rewarded."
A. Fröhlich, Bulletin of the London Mathematical Society, 1978

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Example text

For cp, he observes that, after discarding the factor x and the finitely many factors for which Iwl ~ lxi, logcp can be written as 1 +00 - Le ( n~l -;;- ) (x/w)" ; here the double series consisting of the terms for which Iwl > Ixl and n ~ 3 is absolutely convergent, while has previously been found to be meaningful for the terms corresponding to n = 1 and to n = 2. Incidentally, this amounts to saying that the "Weierstrass canonical product" Le x fI'(1 -~) exp(~ +~), w 2w 2 W which, in the Weierstrass theory, defines the a-function, is absolutely convergent, and then using the earlier results on L;, w - I, L~ w - 2; as these are respectively equal to 0 and to e2 , this gives cp=a·exp( -e 2 x 2 /2).

Is the same as E2 -e 2; moreover, as the trace w +w is a rational integer, w W is contained in W, and so is Nr/w=wr. Applying now the results of Chap. V, §4, to E2 -e 2 and r/w, we see that the left-hand side of (7) is algebraic over Q(e4 ,e6 ); in particular, it is algebraic over Q if e4 and e6 are so. , is homogeneous of degree - 2 in u, v, and of degree 0 in it, V, we can now apply the final results of Chap. (W) is algebraic over Q. § 4. @; clearly these operators commute with each other. For convenience, we shall write ox=%x.

As f is 0 on W, this obviously implies that the values of f, and therefore also those of Et, are algebraic over Q(e4 ,e6 ) at all points of Q W, not in W. Assume now that W has a complex multiplier w; put W' =w W, and take the formula for E! obtained from (5) by differentiation. As E! is homogeneous of degree - 2 in x, U, v, and of degree 0 in it, V, we can write this as follows: t x+r w- 2 IE! (x; W). reR W Subtract x- 2 from both sides and put x=O. , where R' has the usual meaning. Now observe that E!

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