Let E be an elliptic curve defined over a number field K. I prove the following theorems:;Theorem 1 ([FKK12]). If K is a number field containing the cube roots of unity, and E is any elliptic curve defined over K, then there are infinitely many Z/3Z-extensions of K over which E rises in rank. Theorem 2. Let E be an elliptic curve defined over Q with Z/2Z x Z/6Z torsion. Then there are infinitely many cyclic cubic extensions of Q over which E increases in rank.;Theorem 3. If K is a number field containing i and if E is an elliptic curve over K, then there are infinitely many Z/4Z-extensions of K over which E gains in rank.;Theorem 4. Let K be a number field containing the cube roots of unity and let E be any elliptic curve defined over K. Then there are innitely many Z/6Z-extensions of K over which E increases in rank. The first theorem is a result of Fearnley, Kisilevsky, and Kuwata [FKK12] and the second theorem is a special case of another result in the same paper. I give new proofs of these results.
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机译：令E为在数域K上定义的椭圆曲线。我证明以下定理：定理1（[FKK12]）。如果K是一个数字字段，包含1的立方根，并且E是在K上定义的任何椭圆曲线，则K的Z / 3Z扩展无穷多个，E的秩在此扩展。定理2。令E为在Q上定义的椭圆曲线，具有Z / 2Z x Z / 6Z扭转。然后，存在Q的无限多的三次三次三次扩展，E随秩的增加。定理3。如果K是一个包含i的数字字段，并且如果E是在K上的椭圆曲线，则无限大的Z / 4Z扩展定理4。令K为包含1的立方根的数字字段，令E为在K上定义的任何椭圆曲线。然后，K上自然会有许多Z / 6Z扩展，其中E排名上升。第一个定理是Fearnley，Kisilevsky和Kuwata [FKK12]的结果，第二个定理是同一结果中另一个结果的特例。我给出了这些结果的新证据。
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