TY - JOUR

T1 - Horizontal isogeny theorems

AU - Frey, Gerhard

AU - Jarden, Moshe

N1 - Funding Information:
* Research partially supported by the Minkowski Center for Geometry at Tel Aviv University founded by the Minerva Foundation.

PY - 2002

Y1 - 2002

N2 - Let K be a field which is finitely generated over its prime field. Consider elliptic curves E and E′ defined over K. Suppose there exists c ≥: 1 and a set A of prime numbers such that [K(El, El′]: K(El′) ∩ K (El′) ≤, c for all l ∈ Λ. We prove that E′ and E are isogeneous over the algebraic closure of K in each of the following cases: (a) A is infinite and E has no complex multiplication. (b) A is infinite, E has complex multiplication, and char(K) = 0. (c) A has Dirichlet density >3/4, E has complex multiplication, and char(K) > 0.

AB - Let K be a field which is finitely generated over its prime field. Consider elliptic curves E and E′ defined over K. Suppose there exists c ≥: 1 and a set A of prime numbers such that [K(El, El′]: K(El′) ∩ K (El′) ≤, c for all l ∈ Λ. We prove that E′ and E are isogeneous over the algebraic closure of K in each of the following cases: (a) A is infinite and E has no complex multiplication. (b) A is infinite, E has complex multiplication, and char(K) = 0. (c) A has Dirichlet density >3/4, E has complex multiplication, and char(K) > 0.

UR - http://www.scopus.com/inward/record.url?scp=0036033431&partnerID=8YFLogxK

U2 - 10.1515/form.2002.042

DO - 10.1515/form.2002.042

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AN - SCOPUS:0036033431

SN - 0933-7741

VL - 14

SP - 931

EP - 952

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 6

ER -