Almost all elliptic curves are Serre curves

Jones, Nathan

doi:10.1090/s0002-9947-09-04804-1

Cited by 68 publications

(86 citation statements)

References 25 publications

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“…In the cases of the Lang-Trotter conjecture and the Koblitz conjecture, assume that Question 5 has an affirmative answer. Then there exists a constant γ so that We finally use [12,Theorem 4], which in our situation may be stated as follows: This finishes the proof Proposition 24, and thus also the proof of Theorem 6.…”

Section: Lemma 23 For Any Elliptic Curve E Over Q We Havementioning

confidence: 70%

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Averages of elliptic curve constants

Jones¹

2009

Math. Ann.

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Section: Lemma 23 For Any Elliptic Curve E Over Q We Havementioning

confidence: 70%

“…where M E is as in (12). Explicitly, we have Note that when x is an odd integer, our χ 4 (x) coincides with the usual character of conductor 4.…”

Section: The Constants Associated To Serre Curvesmentioning

confidence: 80%

Averages of elliptic curve constants

Jones¹

2009

Math. Ann.

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“…In each case we will show that the corresponding dynamical system D gives rise to a dynamical system D 0 in the space of traces (the trace map) as in diagram (2). The trace map has special geometry: the set of its fixed points (or of periodic points of bounded period) has positive dimension.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Geometry and arithmetic of verbal dynamical systems on simple groups

Bandman¹,

Grunewald²,

Kunyavskiı̆³

et al. 2010

Groups Geom. Dyn.

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Abstract. We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.

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“…In particular, if no prime is not adicallyexceptional for E then A(E) = 1. Moreover, Jones [24] has proved that almost all non-CM elliptic curve over Q have A(E) = 1. On the other hand, if m is a positive integer, A(E) gives necessary conditions to conclude that ρ E,m is non-surjective.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 10. Assuming the Uniformity Conjecture: 3,4,5,6,7,8,9,10,11,12,13,14,15,20,21,24,28,40, 56, 104}.…”

Section: Introductionmentioning

confidence: 99%

Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre's constant associated to E, that gives necessary conditions to conclude that ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m) ≥ valp(A(E)) > 0 then ρE,m is non-surjective. Conditionally under Serre's Uniformity Conjecture, we determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over Q, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of GL2(Zp). Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.Serre's Uniformity Question. If E/Q is a non-CM elliptic curve, then must it be that ρ E,p is surjective for any prime p ≥ 41?Nowadays, an affirmative answer to the above question has received the name of Serre's Uniformity Conjecture (or sometimes just Uniformity Conjecture) despite the fact that Serre himself never conjectured it to be true.

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Almost all elliptic curves are Serre curves

Cited by 68 publications

References 25 publications

Averages of elliptic curve constants

Averages of elliptic curve constants

Geometry and arithmetic of verbal dynamical systems on simple groups

Serre’s Constant of Elliptic Curves Over the Rationals

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