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Title
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SOLUTIONS OF DIOPHANTINE EQUATIONS OVER C(T) AND COMPLEX MULTIPLICATION.
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Identifier
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AAI8312383
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identifier
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8312383
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Creator
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WAJNGURT, CLARA.
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Contributor
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Prof. Harvey Cohn
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Date
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1983
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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In this paper we establish a relationship between the rational solutions (x(t),y(t)), over (//C)(t), of the diophantine equation:;4t('3)x(t)('3)-g(,2)tx(t)-g(,3) =y(t)('2)(4t('3)-g(,2)t-g(,3)), g(,2),g(,3) (ELEM) (1) and the solutions (p(u),p'(u)) which parametrize the elliptic curve E:y('2) = 4x('3)-g(,2)x-g(,3) admitting complex multiplication by (lamda). E is identified with the group (//C)/L whereby L is a lattice which is generated by the periods 2(omega)(,1), 2(omega)(,2), with Im((omega)(,1)/(omega)(,2)) > 0. By definition of complex multiplication by (lamda), we are interested in those multipliers, (lamda) (ELEM) C for which (lamda)L (L-HOOK) L. According to the theory, all such multipliers (lamda) belong to the ring of integers of some imaginary quadratic field K. In particular we restrict out theory to g(,2),g(,3) (ELEM) so that the problem which is discussed here is fully solved for the case of K, having ring class number one. As a result the paper attempts to expand on the work of H. Cohn who dealt with the two specific diophantine equations over (//C)(t), which had specific values for g(,2),g(,3) and which corresponded to the two cases K = (' )(SQRT.(-1), (t = 4p('2)(u)) and K = (' )(SQRT.(-3) (t = 4p('3)(u)). Concluding remarks about solving the problem for K having ring class number greater than one are made.;The basic results of this paper are consequences of results in elliptic function theory having to do with elliptic integrals of the first kind. We first characterize the form of all rational solutions of diophantine equation (1). The rational solutions are derivable from the substitutions.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;in which in which (mu) = 0, (omega)(,1),(omega)(,2),(omega)(,1) + (omega)(,2). As a corollary we show through a minor change in notation, a relationship between the modular invariants of diophantine equation (1) in modified form, and the elliptic curve E, admitting complex multiplication by (lamda). Using techniques established in elliptic function theory, we prove that the complex multiplier (lamda), associated with a unique rational solution (x(t),y(t)), must be of certain form. Next, we construct all rational solutions of diophantine equation (1) by using the addition theorems valid for the Weierstrass function, p(u). When both (lamda),(mu) are non-zero, we use the expression:;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;to find the associated rational solution (x(t),y(t)). Specific examples are worked out for the cases K = (' )(SQRT.(-2) and K = (' )(SQRT.(-7). . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
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Program
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Mathematics
