A bound for the torsion conductor of a non-CM elliptic curve

Jones, Nathan

doi:10.1090/s0002-9939-08-09436-7

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…Proof. Let F E,r (x) be defined as in (5), i.e., F E,r (x) is the main term in (3). The statement of the theorem follows from (2) and the asymptotic estimate…”

Section: Consistency With Chebotarev Densitymentioning

confidence: 95%

“…For convenience, throughout the sequel, we denote the main term on the righthand side of (3) by F E,r (x), i.e., we set (5) F E,r (x) := C E,r x max{2,r 2 /4} Φ E (r/(2 √ t)) 2 √ t log t dt if x ≥ max{2, r 2 /4}. We note that this term is bounded from above by the main term in Conjecture 1, i.e.…”

mentioning

confidence: 99%

“…For convenience, throughout the sequel, we denote the main term on the righthand side of (3) by F E,r (x), i.e., we set (5) F E,r (x) := C E,r x max{2,r 2 /4}…”

mentioning

confidence: 99%

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A Refined Version of the Lang-Trotter Conjecture

Baier

Jones

2008

International Mathematics Research Notices

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Abstract. Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r, where r is a fixed integer. However, as shown in this note, this asymptotic estimate cannot hold for all r in the interval |r| ≤ 2 √ x with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture.The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the "semicircular law", to an asymptotic formula that conjecturally holds for arbitrary integers r in the interval |r| ≤ 2 √ x, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture.

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“…Proof. Let F E,r (x) be defined as in (5), i.e., F E,r (x) is the main term in (3). The statement of the theorem follows from (2) and the asymptotic estimate…”

Section: Consistency With Chebotarev Densitymentioning

confidence: 95%

mentioning

confidence: 99%

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A Refined Version of the Lang-Trotter Conjecture

Baier

Jones

2008

International Mathematics Research Notices

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“…Remark 5.5. The torsion conductor m E of an elliptic curve E over Q is defined in [9] to be the smallest positive integer m ≥ 1 for which…”

Section: Examples and Remarksmentioning

confidence: 99%

$\mathrm{GL}_2$-representations with maximal image

Jones

2015

Mathematical Research Letters

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For a matrix group G, consider a Galois representationwhich extends the cyclotomic character. For a broad class of matrix groups G, we prove a theorem characterizing when such a representation has image which is "as large as possible" inside a fixed open subgroup G ⊆ G( Ẑ). As applications, we obtain such a characterization for the Galois representation on the torsion of a simple principally polarized k-dimensional abelian variety A defined over Q (where G = GSp 2k ) and also for the Galois representation on the torsion of a product of k elliptic curves over Q (whereOur results are motivated by open image theorems for classes of abelian varieties initiated by Serre in the 1960s.

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“…For C E,r in the non-CM case, we reason as follows. According to [9], we may take m E to be of the form For the Koblitz constant in the CM case, we see that…”

Section: Averaging the Serre Curve Constantsmentioning

confidence: 99%