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Efficient Solution of Rational Conics
 Math. Comp
, 1998
"... this paper (section 2), and to Denis Simon for the reference [10]. ..."
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this paper (section 2), and to Denis Simon for the reference [10].
Finiteness theorems for algebraic groups over function fields
, 2010
"... 1.1. Motivation. The most important classes of smooth connected linear algebraic groups G over a field k are semisimple groups, tori, and unipotent groups. The first two classes are unified by the theory of reductive groups, and if k is perfect then an arbitrary G is canonically built up from all th ..."
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1.1. Motivation. The most important classes of smooth connected linear algebraic groups G over a field k are semisimple groups, tori, and unipotent groups. The first two classes are unified by the theory of reductive groups, and if k is perfect then an arbitrary G is canonically built up from all three classes in the sense that
Descents on Curves of Genus 1
, 1995
"... This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material ..."
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This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material is included for which a degree has previously been conferred upon me.
The Arithmetic of Realizable Sequences
, 2003
"... In this thesis we consider sequences of nonnegative integers which arise from counting the periodic points of a map T: X → X, where X is a nonempty set. Some of the main results obtained are concerned with the counting of the periodic points of an endomorphism of a group, in particular when the gr ..."
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In this thesis we consider sequences of nonnegative integers which arise from counting the periodic points of a map T: X → X, where X is a nonempty set. Some of the main results obtained are concerned with the counting of the periodic points of an endomorphism of a group, in particular when the group is locally nilpotent, for which class of groups a localglobal property is established. The ideas developed are applied to some classical sequences, including the Bernoulli and Euler numbers, which are shown to have certain ‘dynamical’ properties. We also consider the LehmerPierce construction for sequences of integers, looking at possible generalizations and their associated measures.
Elliptic curves, rank in families and random matrices
, 2004
"... This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and Lfunctions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some descriptio ..."
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This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and Lfunctions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of MordellWeil groups by means of the Birch and SwinnertonDyer Conjecture; the reader already acquainted with the basics of the theory of elliptic curves can certainly skip it. The second part was originally the writeup of a lecture given for a workshop on the Birch and SwinnertonDyer Conjecture itself, in November 2003 at Princeton University, dealing with what is known and expected about the variation of the rank in families of elliptic curves. Thus it is also a natural continuation of the first part. In comparison with the original text and in accordance with the focus of the first part, more details about the input and confirmations of Random Matrix Theory have been added. Acknowledgments. I would like to thank the organizers of both workshops for
Counterexamples to Hasse Principle: An elementary introduction, http://public.csusm.edu/aitken html/m372/diophantine.pdf
"... Abstract. We give a selfcontained exposition concerning counterexamples to the Hasse principle. Our account, which uses only techniques from undergraduate number theory and algebra, focuses on counterexamples similar to the ..."
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Abstract. We give a selfcontained exposition concerning counterexamples to the Hasse principle. Our account, which uses only techniques from undergraduate number theory and algebra, focuses on counterexamples similar to the
Simple Counterexamples to the Local–Global Principle, at http:// public.csusm.edu/aitken html/m372/diophantine.pdf
"... Abstract. After Hasse had found the first example of a LocalGlobal principle in the 1920s by showing that a quadratic form in n variables represented 0 in rational numbers if and only if it did so in every completion of the rationals, mathematicians investigated whether this principle held in other ..."
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Abstract. After Hasse had found the first example of a LocalGlobal principle in the 1920s by showing that a quadratic form in n variables represented 0 in rational numbers if and only if it did so in every completion of the rationals, mathematicians investigated whether this principle held in other situations. Among the first counterexamples to the Hasse principle were curves of genus 1 constructed by Lind and Reichardt: these were curves without rational points but with points in every completion of Q. In this article we will show that the technique of rational parametrization of conics is powerful enough to derive Reichardt’s result. 1.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Abelian points on algebraic curves
"... Abstract. We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K ab of K. We give: (i) an explicit family of diagonal plane cubic curves without Q abpoints, (ii) for every number field K, a genus one ..."
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Abstract. We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K ab of K. We give: (i) an explicit family of diagonal plane cubic curves without Q abpoints, (ii) for every number field K, a genus one curve C /Q with no K abpoints, and (iii) for every g ≥ 4 an algebraic curve C /Q of genus g with no Q abpoints. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)) abpoints. Convention: All varieties over a field K are assumed to be nonsingular, projective and (as is especially important for what follows) geometrically irreducible. 1.