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One‐parameter families of elliptic curves over ℚ with maximal Galois representations
Author(s) -
Cojocaru AlinaCarmen,
Grant David,
Jones Nathan
Publication year - 2011
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdr001
Subject(s) - mathematics , elliptic curve , sato–tate conjecture , coprime integers , complex multiplication , galois group , integer (computer science) , schoof's algorithm , algebraic number field , division (mathematics) , multiplication (music) , galois module , zero (linguistics) , hessian form of an elliptic curve , supersingular elliptic curve , discrete mathematics , pure mathematics , combinatorics , arithmetic , quarter period , computer science , programming language , linguistics , philosophy
Let E be an elliptic curve over ℚ and let ℚ( E [ n ]) be its n th division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of ℚ( E [ n ])/ℚ is as large as possible, that is, GL 2 (ℤ/ n ℤ), for all integers n coprime to a constant integer m ( E , ℚ) depending (at most) on E /ℚ. Serre also showed that the best one can hope for is to have |GL 2 (ℤ/ n ℤ) : Gal(ℚ( E [ n ])/ℚ)| ⩽ 2 for all positive integers n . We study the frequency of this optimal situation in a one‐parameter family of elliptic curves over ℚ, and show that in essence, for almost all one‐parameter families, almost all elliptic curves have this optimal behavior.