On the field of definition for modularity of CM elliptic curves

Abstract

The purpose of this paper is to decide the conditions under which a CM elliptic curve is modular over its field of definition. r 2004 Elsevier Inc. All rights reserved.

Modularity of CM elliptic curves over division fields

“…Hence all the points of E(C) tor are rational over F , K ab , L = L, K ab , the invariant field of C 1 ∩ C 2 . By Theorem 5.1 in [1], E is modular over L. This completes the proof of Theorem 1.…”

“…Combining Theorem 5.1 in [1] and the second part of Example 3, p. 527 in [3], it follows that E will not be modular over F in general. Therefore, it is important to determine an extension field (a minimal one if possible) of F over which E is modular.…”

“…We fix an isomorphism r : (K ⊗ Q R)/a→E(C), where a is a proper R-ideal in K. The theory of complex multiplication (see Section 3 in [1]) provides a unique homomorphism α E/F from F × A to K × by means of which the action of Gal(F ab /F ) on division points of E may be interpreted class field theoretically. Then L corresponds to…”

“…In the previous paper [1], we gave necessary and sufficient conditions for E to be modular over F , i.e., such a non-zero homomorphism ϕ can be defined over F . Combining Theorem 5.1 in [1] and the second part of Example 3, p. 527 in [3], it follows that E will not be modular over F in general.…”

On construction of certain CM elliptic curves

“…In [2] we proved that E is modular over F (i.e. such a non-zero homomorphism ϕ can be defined over F ) if and only if there exists a Grössen-character γ : group of K) such that γ • N F /K = β E/F , where F is the composite field of F and K in C, N F /K is the norm map from F × A to K × A , and β E/F is the Grössen-character of E/F .…”