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Geometry of numbers, class group statistics and free path lengths
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies.

In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants.

In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.

In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures.

In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. , vi, 13 p.
Series
TRITA-MAT-A, 2015:15
Keyword [en]
geometry of numbers, lattices, class numbers, class groups, free path lengths
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-177888ISBN: 978-91-7595-797-5 (print)OAI: oai:DiVA.org:kth-177888DiVA: diva2:874875
Public defence
2016-01-15, F3, Lindstedsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

QC 20151204

Available from: 2015-12-04 Created: 2015-11-30 Last updated: 2015-12-04Bibliographically approved
List of papers
1. Counting nonsingular matrices with primitive row vectors
Open this publication in new window or tab >>Counting nonsingular matrices with primitive row vectors
2014 (English)In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 173, no 2, 209-230 p.Article in journal (Refereed) Published
Abstract [en]

We give an asymptotic expression for the number of nonsingular integer -matrices with primitive row vectors, determinant , and Euclidean matrix norm less than , as . We also investigate the density of matrices with primitive rows in the space of matrices with determinant , and determine its asymptotics for large .

Keyword
Matrices, Lattices, Primitive vectors, Asymptotics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-142882 (URN)10.1007/s00605-013-0518-x (DOI)000330829000005 ()2-s2.0-84893650451 (Scopus ID)
Note

QC 20140314

Available from: 2014-03-14 Created: 2014-03-13 Last updated: 2017-12-05Bibliographically approved
2. The number of points from a random lattice that lie inside a ball
Open this publication in new window or tab >>The number of points from a random lattice that lie inside a ball
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.

Keyword
geometry of numbers, lattices
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-177287 (URN)
Funder
Swedish Research Council
Note

QC 20160214

Available from: 2015-11-17 Created: 2015-11-17 Last updated: 2016-02-14Bibliographically approved
3. Missing class groups and class number statistics for imaginary quadratic fields
Open this publication in new window or tab >>Missing class groups and class number statistics for imaginary quadratic fields
Show others...
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h increases without bound, and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance the elementary abelian group of order 27 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F(h) for odd h up to 10^6 and F(G) for G a p-group of odd order with |G| up to 10^6. The numerical evidence matches quite well with our conjectures.

Keyword
class groups, class numbers
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-177288 (URN)
Note

QS 2015

Available from: 2015-11-17 Created: 2015-11-17 Last updated: 2015-12-04Bibliographically approved
4. On the free path length distribution for linear motion in an n-dimensional box
Open this publication in new window or tab >>On the free path length distribution for linear motion in an n-dimensional box
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the distribution of free path lengths, or the distance betweenconsecutive bounces of random particles, in an n-dimensional rectangular box.If each particle travels a distance R, then, as R → ∞ the free path lengthscoincides with the distribution of the length of the intersection of a randomline with the box (for a natural ensemble of random lines) and we determinethe mean value of the path lengths. Moreover, we give an explicit formula(piecewise real analytic) for the probability density function in dimension twoand three.In dimension two we also consider a closely related model where eachparticle is allowed to bounce N times, as N → ∞, and give an explicit (againpiecewise real analytic) formula for its probability density function.Further, in both models we can recover the side lengths of the box fromthe location of the discontinuities of the probability density functions.

Keyword
free path lengths
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-177289 (URN)
Note

QS 2015

Available from: 2015-11-17 Created: 2015-11-17 Last updated: 2015-12-04Bibliographically approved

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