A geometric generalization of continued fractions for imaginary quadratic fields

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Kristen Scheckelhoff (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Dan Yasaki

Abstract: The Euclidean Algorithm for the integers is well known and yields a finite continued fraction expansion for each rational number. Geometrically, successive convergents in this expansion correspond to endpoints of edges in the Farey tessellation of the complex upper half plane. The sequence of convergents thus describes a path from the point at infinity to a given rational number, following edges of the tessellation; by identifying points in the upper half plane with positive definite binary quadratic forms (up to scaling), we express this path as a product of matrices in SL2(Z). In general, when the ring of integers of a quadratic number field is Euclidean, there exists a suitable Euclidean function and algorithm with which we may construct a continued fraction expansion for each field element. In these cases, we prove the analogous result that pairs of adjacent convergents determine edges in the Voronoi tessellation of hyperbolic 3-space. We identify points in the upper half space with positive definite binary Hermitian forms (up to scaling), and express the resulting path as a product of matrices in GL2(OF ). Finally, since the Voronoi tessellation exists for all imaginary quadratic fields, including those with non-Euclidean rings of integers, we explore the extent to which this geometric interpretation of continued fractions holds in the general case.

Additional Information

Language: English
Date: 2021
Continued fractions, Hermitian forms, Imaginary quadratic fields, Number theory, Voronoi tessellations
Continued fractions
Quadratic fields
Tessellations (Mathematics)
Generalized spaces

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