On the Siegel–Tatuzawa–Hoffstein theorem

Chen, Yong-Gao

doi:10.4064/aa130-4-5

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…ã ℓ 3n/2 log log ℓ . (7) This certainly implies that d ≥ ℓ for large d, and hence log log d ≥ log log ℓ. Feeding this back into the above estimate gives…”

Section: Proposition 31 (Joshi)mentioning

confidence: 78%

“…Via Dirichlet's class number formula, this comes down to bounding below L(1, −ℓ • ). We will deduce what we need from the following variant of the Siegel-Tatuzawa theorem, due to Chen [7]. Proposition 4.3.…”

Section: Lower Boundmentioning

confidence: 99%

“…If ℓ ≡ 3 (mod 8), Theorem 1.2 shows that the first inequality in (6) can be strengthened by a factor of 3. If ℓ ≡ 7 (mod 8), then −ℓ 2 = 1, and Proposition 3.2 shows that the inequality in (7) can be strengthened by a factor of 3. Following the argument through shows that in either case,…”

Section: 2mentioning

confidence: 99%

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Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:

Bourdon¹,

Pollack²

2016

Int Math Res Notices

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Let GCM(d) denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree d number field. We completely determine GCM(d) for odd integers d and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd d, the set of natural numbers d ′ with GCM(d ′ ) = GCM(d) possesses a well-defined, positive asymptotic density. (2) Let TCM(d) = max G∈G CM (d) #G; under the Generalized Riemann Hypothesis,

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“…ã ℓ 3n/2 log log ℓ . (7) This certainly implies that d ≥ ℓ for large d, and hence log log d ≥ log log ℓ. Feeding this back into the above estimate gives…”

Section: Proposition 31 (Joshi)mentioning

confidence: 78%

Section: Lower Boundmentioning

confidence: 99%

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Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:

Bourdon¹,

Pollack²

2016

Int Math Res Notices

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show abstract

“…Remark 10.7. At present, numerical refinements of Tatuzawa's theorem are available, see [9] and the references therein. In particular, using the main result of [9], one can reduce the numerical bound in Theorem 10.2 to 10 140 .…”

Section: Proof Of Theorem 102mentioning

confidence: 99%