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Zbl 1044.11582
Momose, Fumiyuki
Isogenies of prime degree over number fields. (English)
[J] Compos. Math. 97, No. 3, 329-348 (1995). ISSN 0010-437X; ISSN 1570-5846/e

The author studies isogenies of prime degree between elliptic curves defined over number fields. He proves the following: Let $k$ be a number field. Given an isogeny of degree $p$ between elliptic curves defined over $k$, let $\lambda$ be the character of $\text{Gal}(\overline{k}/k)$ describing the action on its kernel. Then there is an effective constant $C_0$ depending only on $k$ such that for all rational primes $p>C_0$ the character $\lambda$ is of one of three specific types. (Type 3 is expected to consist exactly of isogenies arising from CM elliptic curves.)\par The author also proves that under various hypotheses on $k$ for types 1 and 2 the degree is bounded and for type 3 the isogeny is of CM type.
[Olaf Ninnemann (Berlin)]

MSC 2000:: *11G05 Elliptic curves over global fields

Cited in: Zbl 1216.11060 Zbl 1182.11027

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