On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree

Details On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree

2011-09-28

arxiv.org/abs/1109.6278v2

Mathematics

Number Theory

21 pages. This is a extended version of an earlier version, Sections 3 and 4 are new, Section 2 is from the previous version

Scientific paper

Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree. We also study the following question: for a given $n$ such that $|Y_0(n)(\Q)|>0$, does every elliptic curve over $K$ with an $n$-isogeny have a quadratic twist with torsion $\Z/n\Z$ as a subgroup? We prove that this is true only for the cases $n=2,3,4,6$ and for the pairs $(n,K)=(11, \Q(\zeta_{11})^+)$ and $(n,K)=(14, \Q(\zeta_{7})^+)$.

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