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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. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring's theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre's results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

Inhaltsverzeichnis

Frontmatter

General Theory

Frontmatter

1. Elliptic Functions

Abstract
By a lattice in the complex plane C we shall mean a subgroup which is free of dimension 2 over Z, and which generates C over the reals. If ω1, ω2 is a basis of a lattice L over Z, then we also write L = [ω1, ω2].
Serge Lang

2. Homomorphisms

Abstract
Let A be an elliptic curve defined over a field k. For each positive integer N we denote by AN the kernel of the map
$$t \mapsto \,Nt,\,\,\,\,\,\,\,\,t\, \in \,$$
i.e. it is the subgroup of points of order N.
Serge Lang

3. The Modular Function

Abstract
By SL2 we mean the group of 2 x 2 matrices with determinant 1. We write SL2 (R) for those elements of SL2 having coefficients in a ring R. In practice, the ring R will be Z, Q, R. We call SL2 (Z) the modular group.
Serge Lang

4. Fourier Expansions

Abstract
In this section we derive the promised expansions at infinity for the Gk, whence for Δ and j.
Serge Lang

5. The Modular Equation

Abstract
We are interested in studying the j-invariants of isogenous elliptic curves, which, as we shall see, amounts to studying j ○ α where α is a rational matrix. For this we need some algebraic lemmas concerning integral matrices with positive determinant.
Serge Lang

6. Higher Levels

Abstract
Let Γ = SL2(Z) again. We define ΓN (or Γ(N)) for each positive integer N to be the subgroup of Γ consisting of those matrices satisfying the condition
$$\left({\begin{array}{*{20}{c}}a&b \\c&d \end{array}} \right)\,\, \equiv \,\,1\,\,\,\,(\bmod N),$$
in other words
$$a\,\, \equiv \,\,d\,\, \equiv\,\,1\,\,\,\,(\bmod\,N)\,\,\,\,\,\,\,and\,\,\,\,\,\,\,c\,\, \equiv \,\,b\,\, \equiv \,\,0\,\,\,\,(\bmod \,N).$$
.
Serge Lang

7. Automorphisms of the Modular Function Field

Abstract
If N, M are positive integers, and N|M, then we have a canonical homomorphism
$$G{L_2}\left( {Z/MZ} \right)\,\, \to \,\,G{L_2}\left( {Z/NZ} \right),$$
and we can take the projective limit.
Serge Lang

Complex Multiplication Elliptic Curves with Singular Invariants

Frontmatter

8. Results from Algebraic Number Theory

Abstract
In this chapter we assume that the reader is acquainted with the ordinary ideal theory in number fields. Cf. for instance [B7]. The first two sections should be read as technical background for Chapter 10, §2. On the other hand, although we strive for some completeness, once the reader sees the first results that the proper o-lattices form a multiplicative group, he can wait to read the other results until he needs them, as they are slightly technical. They are all classical, known to Dedekind, except possibly for the fact that a proper o-lattice is locally principal, which seems to have been first pointed out by Ihara [26]. The localization technique will be used heavily for the idelic formulation of the complex multiplication, as in Shimura [B12].
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9. Reduction of Elliptic Curves

Abstract
The properties of reduction in this chapter, except for §3, are due to Deuring, who used them to give his algebraic proofs for complex multiplication. We shall not give any proofs. These can be given ad hoc, as Deuring did, for the elliptic curves, or one can develop a general reduction theory, as in Shimura [39]. No matter what, it is a pain to lay these foundations, but the results can be stated simply. Although classically one reduces over a discrete valuation ring, it is useful to deal with an arbitrary local ring.
Serge Lang

10. Complex Multiplication

Abstract
We first consider values of the j-function at quadratic imaginary numbers. We shall see that these values generate abelian extensions of quadratic fields.
Serge Lang

11. Shimura’s Reciprocity Law

Abstract
Let F be the modular function field, studied in Chapter 6. We saw that F can be identified with the field of x-coordinates (or h-coordinates, h = Weber function) of division points of an elliptic curve A defined over Q(j), having invariant j. Let k be an imaginary quadratic field, and let zk ⋂ ℌ
Serge Lang

12. The Function Δ(ατ)/Δ(τ)

Abstract
In this section we give an example for the Shimura theorem concerning the quotient of automorphic functions.
Serge Lang

13. The ι-adic and p-adic Representations of Deuring

Abstract
It was first proved by Hasse that even in characteristic p > 0, if N is an integer prime to p, then the points of order N on an elliptic curve A form a cyclic group of type Z/NZ x Z/NZ.
Serge Lang

14. Ihara’s Theory

Abstract
One can reduce the modular function field mod p and obtain an infinite extension of F p (j), with j transcendental over F p . Igusa determined the Galois group [22], pointing out that it has the sameSL2 part as in characteristic zero, and that the part acting on the roots of unity is just that generated by the Frobenius element, i.e. those matrices having determinant a power of p. Ihara had the idea of lifting back singular values \(bar j\) of j in the algebraic closure a F p by the Deuring lifting, and to represent the Frobenius automorphism in the decomposition group of the modular function field in characteristic p by an element of the isotropy group of the point z ∈ ℌ such that \(overline {j(z)}=\overline j\), with a suitable place of the algebraic numbers, denoted also by a bar. This led him to deep conjectures concerning non-abelian extensions of the rational field F p (j), for which we refer to his original treatise [B6].
Serge Lang

Elliptic Curves with Non-Integral Invariants

Frontmatter

15. The Tate Parametrization

Abstract
In this section, we have essentially copied an unpublished manuscript of Tate. For an exposition of Tate’s results which is more complete we refer to Roquette [B9]. We have done essentially what is needed to prove the isogeny theorem afterwards.
Serge Lang

16. The Isogeny Theorems

Abstract
We return to p-adic representations. Let A be an elliptic curve defined over K. We take points of A in a fixed algebraic closure Ka. We have the p-adic spaces T p (A) and V p (A) over Z p and Q p respectively.
Serge Lang

17. Division Points over Number Fields

Abstract
We know from Chapter 2, §1 that over any field K, the Galois group of the field obtained by adjoining to K all coordinates of points of finite order on an elliptic curve A defined over K is representable as a closed subgroup of the product
$$\prod\limits_{\ell } {G{{L}_{2}}\left( {{{Z}_{\ell }}} \right)} , $$
taken over primes ℓ.
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Theta Functions and Kronecker Limit Formulas

Frontmatter

18. Product Expansions

Abstract
Both in number theory and analysis one factorizes elements into prime powers. In analysis, this means that a function gets factored into an infinite product corresponding to its zeros and poles. Taking the values at special points, such an analytic expression reflects itself into special properties of the values, for which it becomes possible to determine the prime factorization in number fields.
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19. The Siegel Functions and Klein Forms

Abstract
This chapter is entirely rewritten for the second edition, and follows Kubert-Lang (Units in the Modular Function Field I, Math. Ann, 218 (1975), pp. 67–96; see also “Modular Units”, Springer-Verlag, 1981).
Serge Lang

20. The Kronecker Limit Formulas

Abstract
Let f be a function on R. We shall say that f tends to 0 rapidly at infinity if for each positive integer m the function
$$x \mapsto {\left| x \right|^m}f(x)$$
is bounded. We define the Schwartz space S to be the set of functions on R which are infinitely differentiable and which tend to 0 rapidly at infinity, as well as their derivatives of all orders.
Serge Lang

21. The First Limit Formula and L-series

Abstract
Let k be an imaginary quadratic field, with discriminant —dk < 0 so that dk is the absolute value of the discriminant. Let o be the ring of integers in k, and let a be an ideal class.
Serge Lang

22. The Second Limit Formula and L-series

Abstract
Let k be a number field and o = ok the ring of algebraic integers. Let f be an ideal of o. (Unless otherwise specified, ideal means contained in o.) We shall consider Gauss sums formed with characters (the generalization to number fields is due to Hecke).
Serge Lang

Backmatter

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