椭圆函数 第2版（影印版）

　　 Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.
　　 Some new techniques and outlooks have recently appeared on these old subjects, continuing in the tradition of Kronecker, Weber, Fricke, Hasse,Deuring. Shimura's book Introduction to the arithmetic theory of automorphic functions is a splendid modern reference, which I found very helpful myself to learn some aspects of elliptic curves. It emphasizes the direction of the Hasse-Weil zeta function, Hecke operators, and the generalizations due to him to the higher dimensional case (abelian varieties, curves of higher genus coming from an arithmetic group operating on the upper half plane, bounded symmetric domains with a discrete arithmetic group whose quotient is algebraic). I refer the interested reader to his book and the bibliography therein.
　　

part one general theory
chapter 1 elliptic functions
1 the liouville theorems
2 the weierstrass function
3 the addition theorem
4 isomorphism classes of elliptic curves
5 endomorphisms and automorphisms
chapter 2 homomorphisms
1 points of finite order
2 isogenies
3 the involution
chapter 3 the modular function
1 the modular group
2 automorphic functions of degree 2k
3 the modular functionj
chapter 4 fourier expansions
1 expansion for gk, g2, g3, △ and j
2 expansion for the weierstrass function
3 bernoulli numbers
chapter 5 the modular equation

　　 Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.
　　 Some new techniques and outlooks have recently appeared on these old subjects, continuing in the tradition of Kronecker, Weber, Fricke, Hasse,Deuring. Shimura's book Introduction to the arithmetic theory of automorphic functions is a splendid modern reference, which I found very helpful myself to learn some aspects of elliptic curves. It emphasizes the direction of the Hasse-Weil zeta function, Hecke operators, and the generalizations due to him to the higher dimensional case (abelian varieties, curves of higher genus coming from an arithmetic group operating on the upper half plane, bounded symmetric domains with a discrete arithmetic group whose quotient is algebraic). I refer the interested reader to his book and the bibliography therein.
　　 I have placed a somewhat different emphasis in the present exposition. First,I assume less of the reader, and start the theory of elliptic functions from scratch. I do not discuss Hecke operators, but include several topics not covered by Shimura, notably the Deuring theory of l-adic and p-adic representations;the application to Ihara's work; a discussion of elliptic curves with non-integral invariant, and the Tate parametrization, with the applications to Serre's work on the Galois group of the division points over number fields, and to the isogeny
　　theorem; and finally the Kronecker limit formula and the discussion of values of special modular functions constructed as quotients of theta functions, which are better than values of the Weierstrass function because they are units when properly normalized, and behave in a specially good way with respect to the action of the Galois group.
　　 Thus the present book has a very different flavor from Shimura's. It was unavoidable that there should be some non-empty overlapping, and I have chosen to redo the complex multiplication theory, following Deuring's algebraic method, and reproducing some of Shimura's contributions in this line (with some simplifications, e.g. to his reciprocity law at fixed points, and with another proof for the theorem concerning the automorphisms of the modular function field).
　　 I do not emphasize elliptic curves in characteristic p, except as they arise by reduction from characteristic O. Thus I have omitted most of the theory proper to characteristic p, especially the finer theory of supersingutar invariants. The reader should be warned, however, that this theory is important for the deeper analysis of the arithmetic theory of elliptic curves. The two appendices should help the reader get into the literature.
　　 I thank Shimura for his patience in explaining to me some facts about his research; Eli Donkar for his notes of a course which provided the basis for the present book; Swinnerton-Dyer and Walter Hill for their careful reading of the manuscript.
　　New Haven, Connecticut SERGE LANG
　　 Note for the Second Edition
　　 I thank Springer-Verlag for keeping the book in print. It is unchanged except for the corrections of some misprints, and two items:
　　 1. John Coates pointed out to me a mistake in Chapter 21, dealing with the L-functions for an order. Hence I have eliminated the reference to orders at that point, and deal only with the absolute class group.
　　 2. I have renormalized the functions in Chapter 19, following Kubert-Lang.Thus I use the Klein forms and Siegel functions as in that reference. Actually, the final formulation of Kronecker's Second Limit Formula comes out neater under this renormalization.
　　 S.L.
　　 November 1986