Indivisibility of class numbers of imaginary quadratic function fields

Byeon, Dongho

doi:10.4064/aa132-4-6

Cited by 3 publications

(3 citation statements)

References 8 publications

(11 reference statements)

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“…In particular, for q > Q 0 (ℓ), a positive fraction of imaginary quadratic extensions of F q (t) have class number divisible by ℓ, and a positive fraction have class number indivisible by ℓ. The infinitude of quadratic extensions of F q (t) with class number divisible by ℓ was previously known ( [18], [10]) as was the corresponding result for indivisibility by ℓ [29], but in both cases without a positive proportion.…”

Section: Introductionmentioning

confidence: 67%

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

2016

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Abstract. We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let ℓ > 2 be prime and A a finite abelian ℓ-group. Then there exists Q = Q(A) such that, for q greater than Q, a positive fraction of quadratic extensions of Fq(t) have the ℓ-part of their class group isomorphic to A.

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Section: Introductionmentioning

confidence: 67%

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

2016

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“…Remark 2.2. The weight 3/2 modular form ∞ n=0 r(n)q n := Θ(z) 3 is intimately tied to class numbers for imaginary quadratic fields. It is well known that the r(n) are given by Hurwitz class numbers H(n).…”

Section: Hurwitz Mock Modular Formsmentioning

confidence: 99%

“…Vatsal [32] used a theorem of Gross and Zagier [12] to show that the elliptic curve E = X 0 (19) has M r E (X) ≫ X for r = 0, 1. Vatsal's argument was extended by Byeon [3] to elliptic curves in the isogeny class of an elliptic curve with a nontrivial cuspidal 3-torsion point and square-free conductor.…”

Section: Introductionmentioning

confidence: 99%

Indivisibility of class numbers of imaginary quadratic fields

Beckwith

2017

Res Math Sci

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We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to −X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen-Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves. BackgroundIdeal class numbers of imaginary quadratic fields have been studied since Gauss, who conjectured that for any given h, there are only finitely many negative fundamental discriminants D such that h(D) = h. The history of Gauss' Conjecture is rich. The conjecture was shown to be true by work of Heilbronn [13], who did not show how to find the imaginary quadratic fields with a given class number. Siegel [26] proved that h(−D) grows like |D| 1/2 , but did so ineffectively. In other words, for each > 0 he proved that for sufficiently large D there are positive constants c 1 and c 2 for which

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Indivisibility of divisor class numbers of Kummer extensions over the rational function field

Lee

Yoo

2018

Journal of Number Theory

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Indivisibility of class numbers of imaginary quadratic function fields

Cited by 3 publications

References 8 publications

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Indivisibility of class numbers of imaginary quadratic fields

Indivisibility of divisor class numbers of Kummer extensions over the rational function field

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