Complex Multiplication: Kronecker’s Jugendtraum for Q(i)
نویسنده
چکیده
We discuss Chapter 6 from Silverman & Tate’s Rational Points on Elliptic Curves, in which the authors outline a means of proving Kronecker’s Jugendtraum for Q and Q(i). One considers C[n], the kernel of the multiplication-by-n map on an elliptic curve C. After defining the number field Q(C[n]), we examine instances in which its Galois group over a field K is abelian. For the elliptic curve
منابع مشابه
Kronecker’s Jugendtraum
We give a brief introduction to Kronecker’s Jugendtraum, sometimes called Hilbert’s twelfth problem, which concerns ways to describe abelian extensions of arbitrary number field K using special values of transcendental functions. Only partial results have been proved and the general case still remains open until now. Specifically, we will explain how the problem is solved when K is an imaginary...
متن کاملHeegner points, Stark-Heegner points, and values of L-series
Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...
متن کاملArithmetic and Attractors
We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the “attractor mechanism” of N = 2 supergravity. In IIB string compactification this mechanism singles out certain “attractor varieties.” We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N = 4, 8 compactificat...
متن کاملASPECTS OF COMPLEX MULTIPLICATION Contents
1. Preview 2 Complex multiplication on elliptic curves over C 2 Traces of singular moduli 3 Class field theory 3 The Kronecker limit formula and Kronecker’s solution of Pell’s equation 4 Application to Diophantine equations (Villegas) 4 L-series and CM modular forms 5 Other topics 6 2. Complex Multiplication on Elliptic Curves over C 6 Elliptic Curves over C 6 Elliptic functions 7 Complex multi...
متن کاملModuli Fields of Cm-motives Applied to Hilbert's 12-th Problem
We apply the theorem of Shimura and Taniyama to compute the largest abelian extension of a CM-eld E which can be obtained by evaluating E-rational automorphic functions at certain special points of Shimura varieties. Abelian class eld theory classiies the abelian extensions of a number eld k, but does not explain how to generate the elds. In his Jugendtraum, Kronecker suggested that the abelian...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008