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 -