Random matrices, the Cohen–Lenstra heuristics, and roots of unity

Garton, Derek

doi:10.2140/ant.2015.9.149

Cited by 25 publications

(33 citation statements)

References 17 publications

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“…They show that the mod p Selmer group actually does arise as the intersection of two maximal isotropics -the question, then, is whether these isotropics are in fact 'uniformly distributed' in an appropriate sense. This distribution also appears in the conjectures of Malle [16] and Garton [13] as the conjectured distribution of p-ranks of ideal class groups of number fields containing pth roots of unity. (3) The conjectured distribution of the a-number is the same as the distribution on the dimension of the fixed space of a random large symplectic matrix over F q , which was computed in an unpublished work by Rudvalis and Shinoda [24] (see [11] for a review of their results and an alternative proof).…”

Section: Statistics Of Random Dieudonné Modulesmentioning

confidence: 69%

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Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Cais¹,

Ellenberg²,

Zureick-Brown

2013

J. Inst. Math. Jussieu

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Link to this article: http://journals.cambridge.org/abstract_S1474748012000862How to cite this article: Bryden Cais, Jordan S. Ellenberg and David Zureick-Brown (2013). Random Dieudonné modules, random -divisible groups, and random curves over nite elds.Abstract We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as a 'uniform distribution', and we compute the distribution of various statistics (p-corank, a-number, etc.) of p-divisible groups drawn from this distribution. It is then natural to ask to what extent the p-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over k are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen-Lenstra type for char k = p, in which case the random p-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F 3 appear substantially less likely to be ordinary than hyperelliptic curves over F 3 .

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Section: Statistics Of Random Dieudonné Modulesmentioning

confidence: 69%

“…(This point of view begins with Friedman and Washington [12] and has subsequently been refined by Achter [1], Malle [16], and Garton [13].) (This point of view begins with Friedman and Washington [12] and has subsequently been refined by Achter [1], Malle [16], and Garton [13].)…”

Section: Introductionmentioning

confidence: 99%

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

Cais¹,

Ellenberg²,

Zureick-Brown

2013

J. Inst. Math. Jussieu

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“…Remark 7.7. There are other predictions for Prob(rk 2 C K = ρ): one due to Venkatesh-Ellenberg involving the Schur multiplier [47, § 2.4], work of Garton [24] accounting for roots of unity, and theorems of Wood in the function field case [50] that suggest predictions in the number field case. It is possible that these different perspectives all agree with Conjecture 7.3, but this has not yet been established.…”

Section: Conjectures: 2-ranks Of Narrow Class Groupsmentioning

confidence: 99%

The 2‐Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

Dummit

Voight

2018

Proc. London Math. Soc.

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We investigate in detail a homomorphism which we call the 2-Selmer signature map from the 2-Selmer group of a number field K to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields K with given narrow class group 2-rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an Appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces. Contents

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“…They then conjectured that if A ∈ G, then ν(A) is the probability that the ℓ-Sylow subgroup of the ideal class group of an imaginary quadratic number field is isomorphic to A. Since then, mathematicians have defined various probability distributions on G and conjectured that these distributions describe various phenomena, both number-theoretic (e.g., [FW89], [CM90], [EVW09], [Mal10], [Gar15]) and combinatorial (e.g., [Mat14], [CKL + ]). Given any discrete probability distribution ξ ∶ G → R ≥0 and any A ∈ G, define the Ath moment of ξ to be B∈G Surj (B, A) ξ(B),…”

Section: Introductionmentioning

confidence: 99%

“…A precise description of these "favorable conditions", however, is still elusive. In [EVW09], [Mat14], and [Gar15], for example, the moments of the particular discrete probability distributions on G in question completely determine the distribution. In [Gar15], a Möbius inversion-type procedure transforms closed formulas for moments of certain distributions on G into closed formulas for the distribution itself.…”

Section: Introductionmentioning

confidence: 99%