By Andre Weil

ISBN-10: 3540650369

ISBN-13: 9783540650362

ISBN-10: 3642662099

ISBN-13: 9783642662096

"As a contribution to the background of arithmetic, this can be a version of its style. whereas adhering to the elemental outlook of Eisenstein and Kronecker, it offers new perception into their paintings within the mild of next advancements, correct as much as the current day. As one could anticipate from this writer, it additionally comprises a few pertinent reviews taking a look into the long run. it's not even though 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 publication has its personal very person flavour, reflecting a kind of mixed Eisenstein-Kronecker-Weil character. dependent basically on Eisenstein's method of elliptic services through endless sequence over lattices within the advanced aircraft, it stretches again to the very beginnings at the one hand and reaches ahead to a few of the newest learn paintings at the different. (...) The continual reader can be richly rewarded."*A. Fröhlich, Bulletin of the London Mathematical Society, 1978*

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**Example text**

The theory of distributions allows a re-interpretation of some of the above results; this will now be sketched briefly. As is well-known, there is, on R, on C and more generally on every local field, one and (up to a constant factor) only one distribution which, under the action of the multiplicative group of that field, gets multiplied by a given quasicharacter W of the latter group; such a distribution (which, suitably normalized, is sometimes known as "the Tate distribution") will be said to belong to w.

One finds then that the function F(q)P(q) is unchanged if one substitutes q4 for q; as it is a power-series in q, beginning with the constant term 1 and convergent for Iql < 1, it must be 1. § 7. It seems more interesting, however, to derive the same result from the partial differential equations of parabolic type for E1 and for T. ActuaHy (9) of§2 is such an equation, since oEt/ox=-E2 , oE2 /ox=-2E3 ; and the fact that a "theta-series" such as T(q,z) satisfies a parabolic equation is well-known; even before Jacobi, theta-series had been introduced into the mathematical literature by Fourier in his work on the heat equation.

The attempt will not be made here to disentangle all these threads. In this chapter we will merely treat the simple series as a preparation for our description of Kronecker's work on the double series of type (2). § 4. In this chapter we denote by X a character of the group Z; we will usually write it as with yeR; whenever convenient, one may assume series (4) Sa(x,y,s) = L*(x + ,unx+ ,u1-2s e( - O~y< 1. We consider the ,uy), I' where a is an integer ~ 0, y is real, x and s are complex, and where L* denotes the sum taken over all integers ,u =l= - x (i.

### Elliptic Functions according to Eisenstein and Kronecker by Andre Weil

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