$\ell $-torsion in class groups of certain families of $D_4$-quartic fields

An, Chen

doi:10.5802/jtnb.1109

Cited by 6 publications

(1 citation statement)

References 11 publications

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“…(The families of fields are too numerous to describe here, but include for example: totally ramified cyclic extensions of Q, in which case the result is unconditional; degree n S n -fields with squarefree discriminant, conditional for n ≥ 5 on the strong Artin conjecture and certain counts for number fields; degree n A n -fields, conditional on the strong Artin conjecture.) We note that this approach has recently been adapted by An [An18] to certain families of D 4 -quartic fields. We note that an interesting contrast of [PTBW20] to [EPW17] is that while it also depends heavily on counting number fields, it only requires rough counts.…”

Section: Average and "Almost All" Resultsmentioning

confidence: 99%

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Pierce

Turnage-Butterbaugh

Wood

2021

Mathematical Research Letters

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It is conjectured that within the class group of any number field, for every integer ℓ ≥ 1, the ℓ-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the ℓ-torsion conjecture has relied crucially on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the ℓ-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the ℓ-torsion conjecture to be true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments." Pierce, Turnage-Butterbaugh, and Wood 7 Known results toward Conjecture 1.1, and relation to GRH 606 8 Appendices 611 References 614 1/2+ε K , 1 We will use Vinogradov's notation: A ≪ B denotes that there exists a constant C such that |A| ≤ CB, and A ≪ κ B denotes that C may depend on κ.

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Section: Average and "Almost All" Resultsmentioning

confidence: 99%

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Pierce

Turnage-Butterbaugh

Wood

2021

Mathematical Research Letters

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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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Log-free zero density estimates for automorphic L-functions

2024

Journal of Number Theory

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$\ell $-torsion in class groups of certain families of $D_4$-quartic fields

Cited by 6 publications

References 11 publications

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

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

Log-free zero density estimates for automorphic L-functions

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