Progress towards counting<i>D</i><sub>5</sub>quintic fields

Larson, Eric; Rolen, Larry

doi:10.2140/involve.2012.5.91

Cited by 2 publications

(4 citation statements)

References 8 publications

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“…The exponent 0.7 + ε of Cohen and Thorne is just slightly above the latter. Our bound is X 0.678... , and hence is slightly better than the bound suggested by the numerical experiments in [28].…”

Section: Corollary 12 Let P Be An Odd Prime and ε > 0 Then We Havecontrasting

confidence: 46%

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Averages and higher moments for the $$\ell $$-torsion in class groups

Frei

Widmer

2020

Math. Ann.

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We prove upper bounds for the average size of the $$\ell $$ ℓ -torsion $${{\,\mathrm{Cl}\,}}_K[\ell ]$$ Cl K [ ℓ ] of the class group of K, as K runs through certain natural families of number fields and $$\ell $$ ℓ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for $${{\,\mathrm{Cl}\,}}_K[\ell ]$$ Cl K [ ℓ ] to the existence of many primes splitting completely in K that are small compared to the discriminant of K. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of $${{\,\mathrm{Cl}\,}}_K[\ell ]$$ Cl K [ ℓ ] for many families of number fields K considered in the literature, for example, for the families of all degree-d-fields for $$d\in \{2,3,4,5\}$$ d ∈ { 2 , 3 , 4 , 5 } (and non-$$D_4$$ D 4 if $$d=4$$ d = 4 ). As an application of the case $$d=2$$ d = 2 we obtain the best upper bounds for the number of $$D_p$$ D p -fields of bounded discriminant, for primes $$p>3$$ p > 3 .

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Section: Corollary 12 Let P Be An Odd Prime and ε > 0 Then We Havecontrasting

confidence: 46%

“…The special case p = 5 was also considered by Larsen and Rolen [28]. They suggest to improve Klüners' bound X 0.75+ε [25,Theorem 2.7] by counting integral points on a variety defined by a norm equation.…”

Section: Corollary 12 Let P Be An Odd Prime and ε > 0 Then We Havementioning

confidence: 99%

Averages and higher moments for the $$\ell $$-torsion in class groups

Frei

Widmer

2020

Math. Ann.

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show abstract

Section: Introductioncontrasting

confidence: 46%

“…The special case p = 5 was also considered by Larsen and Rolen [LR12]. They suggest to improve Klüners' bound X 0.75+ε [Klü06, Theorem 2.7] by counting integral points on a variety defined by a norm equation.…”

Section: Introductionmentioning

confidence: 99%

Averages and higher moments for the $\ell$-torsion in class groups

Frei¹,

Widmer²

2018

Preprint

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We prove upper bounds for the average size of the ℓ-torsion Cl K [ℓ] of the class group of K, as K runs through certain natural families of number fields and ℓ is a positive integer. We refine a key argument, used in almost all results of this type, which links upper bounds for Cl K [ℓ] to the existence of many primes splitting completely in K that are small compared to the discriminant of K. Our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument, in conjunction with new counting results for these invariants. This leads to significantly improved upper bounds for the average and sometimes even higher moments of Cl K [ℓ] for many families of number fields K considered in the literature, for example, for the families of all degree-d-fields for d ∈ {2, 3, 4, 5} (and non-D 4 if d = 4). As an application of the case d = 2 we obtain the best upper bounds for the number of D p -fields of bounded discriminant, for primes p > 3.

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Progress towards countingD₅quintic fields

Cited by 2 publications

References 8 publications

Averages and higher moments for the $$\ell $$-torsion in class groups

Averages and higher moments for the $$\ell $$-torsion in class groups

Averages and higher moments for the $\ell$-torsion in class groups

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Progress towards countingD5quintic fields

Cited by 2 publications

References 8 publications

Averages and higher moments for the $$\ell $$-torsion in class groups

Averages and higher moments for the $$\ell $$-torsion in class groups

Averages and higher moments for the $\ell$-torsion in class groups

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Progress towards countingD₅quintic fields