Sums of squares in function fields of elliptic curves

http://hdl.handle.net/10150/289166
Title:
Sums of squares in function fields of elliptic curves
Author:
Cunningham, Geoffrey William
Issue Date:
2000
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In this dissertation we examine the problem of determining restricted representation numbers. Let k be the function field of a curve over a finite field and let w be a place of k of degree 1. Then r₂ᵢ(ξ,n) denotes the number of representations of ξ as a sum of 2Ι squares where the summands are integral away from w and have a pole of order at most n at w. This problem has been studied by Merrill and Walling in the case of the rational function field, Fq(T), by relating the restricted representation numbers to the Fourier coefficients of the 2Ι-th power of a theta function. In this dissertation we are interested in the case where k is the function field of an elliptic curve over a finite field of order q, a power of a prime. We use a more general theta function defined by Weil. A transformation law or functional equation for this theta function follows almost immediately from Weil's results. The analogue of the classical inversion formula is a special case of this transformation law. In the case of the rational function field we show that Weil's theta function specializes to the theta function used by Merrill and Walling. In the elliptic curve case, the restricted representation numbers are simply a factor of qⁿˡ times the Fourier coefficients of the 2Ι-th power of Weil's theta function. Thus, we may simply study the Fourier coefficients. We obtain a formula for these Fourier coefficients via classical techniques and then derive a recurrence relation for the Fourier coefficients by applying the transformation law to the formula. We are unable to solve the recurrence relation completely. However, a careful analysis of the recurrence relation and of the values of the theta function itself, along with some lemmas on character sums, lead to asymptotic formulas. We show that the Fourier coefficients are asymptotic to qⁿ⁽ˡ⁻²⁾ as Ι → ∞ and to a constant times qⁿ⁽ˡ⁻²⁾ as n → ∞.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Ulmer, Douglas

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleSums of squares in function fields of elliptic curvesen_US
dc.creatorCunningham, Geoffrey Williamen_US
dc.contributor.authorCunningham, Geoffrey Williamen_US
dc.date.issued2000en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn this dissertation we examine the problem of determining restricted representation numbers. Let k be the function field of a curve over a finite field and let w be a place of k of degree 1. Then r₂ᵢ(ξ,n) denotes the number of representations of ξ as a sum of 2Ι squares where the summands are integral away from w and have a pole of order at most n at w. This problem has been studied by Merrill and Walling in the case of the rational function field, Fq(T), by relating the restricted representation numbers to the Fourier coefficients of the 2Ι-th power of a theta function. In this dissertation we are interested in the case where k is the function field of an elliptic curve over a finite field of order q, a power of a prime. We use a more general theta function defined by Weil. A transformation law or functional equation for this theta function follows almost immediately from Weil's results. The analogue of the classical inversion formula is a special case of this transformation law. In the case of the rational function field we show that Weil's theta function specializes to the theta function used by Merrill and Walling. In the elliptic curve case, the restricted representation numbers are simply a factor of qⁿˡ times the Fourier coefficients of the 2Ι-th power of Weil's theta function. Thus, we may simply study the Fourier coefficients. We obtain a formula for these Fourier coefficients via classical techniques and then derive a recurrence relation for the Fourier coefficients by applying the transformation law to the formula. We are unable to solve the recurrence relation completely. However, a careful analysis of the recurrence relation and of the values of the theta function itself, along with some lemmas on character sums, lead to asymptotic formulas. We show that the Fourier coefficients are asymptotic to qⁿ⁽ˡ⁻²⁾ as Ι → ∞ and to a constant times qⁿ⁽ˡ⁻²⁾ as n → ∞.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US