Counting elliptic curves with an isogeny of degree three
نویسندگان
چکیده
منابع مشابه
On Elliptic Curves with an Isogeny of Degree
We show that if E is an elliptic curve over Q with a Q-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to E is as large as allowed by the isogeny, except for the curves with complex multiplication by Q( √ −7). The analogous result with 7 replaced by a prime p > 7 was proved by the first author in [8]. The present case p = 7 has additional interesting co...
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giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of ...
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Mazur’s theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if G is one of these 15 groups, we show that the number of elliptic curves up to height X whose torsion subgroup is isomorphic to G is on the order of X, for some number d = d(G) which we c...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society, Series B
سال: 2020
ISSN: 2330-1511
DOI: 10.1090/bproc/45