Unformatted text preview: 105 Remark 10.2 . Notice that τ ∈ Q ( √- d ) if and only if the lattice Λ τ has complex multiplication (see Lecture 2). By Exercise 2.6 this is equivalent to that E τ 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 πiz/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 ”J¨ugendtraum” of Leopoldwith abelian Galois group....
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
- Fall '09