dc.contributor.advisor Ulmer, Douglas en_US dc.contributor.author Cunningham, Geoffrey William dc.creator Cunningham, Geoffrey William en_US dc.date.accessioned 2013-05-09T09:37:10Z dc.date.available 2013-05-09T09:37:10Z dc.date.issued 2000 en_US dc.identifier.uri http://hdl.handle.net/10150/289166 dc.description.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 → ∞. dc.language.iso en_US en_US dc.publisher The University of Arizona. en_US dc.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. en_US dc.subject Mathematics. en_US dc.title Sums of squares in function fields of elliptic curves en_US dc.type text en_US dc.type Dissertation-Reproduction (electronic) en_US thesis.degree.grantor University of Arizona en_US thesis.degree.level doctoral en_US dc.identifier.proquest 9983868 en_US thesis.degree.discipline Graduate College en_US thesis.degree.discipline Mathematics en_US thesis.degree.name Ph.D. en_US dc.identifier.bibrecord .b40823489 en_US refterms.dateFOA 2018-05-18T00:40:53Z html.description.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 → ∞.
﻿

### Files in this item

Name:
azu_td_9983868_sip1_m.pdf
Size:
2.371Mb
Format:
PDF