MODULAR FORMS-page109

MODULAR FORMS-page109 - 105 Remark 10.2 . Notice that Q ( -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 105 Remark 10.2 . Notice that Q ( - d ) if and only if the lattice has complex multiplication (see Lecture 2). By Exercise 2.6 this is equivalent to that E has endomorphism ring larger than Z . An elliptic curve with this property is called an elliptic curve with complex multiplication . Viewing j as a function on the set of isomorphism classes of elliptic curves, the previous corrollary says that the value of j at the isomorphism class of an elliptic curve with complex multiplication is an algebraic integer. Remark 10.3 . The classical Kronecker Theorem asserts that any finite abelian exten- sion of Q with abelian Galois group can be obtained by joining roots of unity to Q . Observe that a n th root of unity is the value of the function f ( z ) = e 2 iz/n on Z . Let K be an imaginary quadratic extension of Q and let a be an ideal in the ring of integers of K . Then the set j ( a ) generates a maximal non-ramified extension of the field K with abelian Galois group. This is the celebrated Jugendtraum of Leopoldwith abelian Galois group....
View Full Document

Ask a homework question - tutors are online