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Lattice methods for finding rational points on varieties over number fields
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Turner, Charlotte L. (2013) Lattice methods for finding rational points on varieties over number fields. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2715884~S1
Abstract
We develop a method for finding all rational points of bounded height on a variety
defined over a number field K. Given a projective variety V we find a prime p
of good reduction for V with certain properties and find all points on the reduced
curve V (Fp). For each point P 2 V (Fp) we may define lattices of lifts of P: these
lattices contain all points which are congruent to P mod p satisfying the defining
polynomials of V modulo a power of p. Short vectors in these lattices are possible
representatives for points of bounded height on the original variety V (K). We make
explicit the relationship between the length of a vector and the height of a point
in this setting. We will discuss methods for finding points in these lattices and
how they may be used to find points of V (K), including a method involving lattice
reduction over number fields.
The method is implemented in Sage and examples are included in this thesis.
Item Type: | Thesis (PhD) |
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Subjects: | Q Science > QA Mathematics |
Library of Congress Subject Headings (LCSH): | Lattice theory, Algebraic fields, Rational points (Geometry), Algebraic varieties |
Official Date: | December 2013 |
Institution: | University of Warwick |
Theses Department: | Mathematics Institute |
Thesis Type: | PhD |
Publication Status: | Unpublished |
Supervisor(s)/Advisor: | Cremona, J. E. |
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) |
Extent: | vii, 98 leaves. |
Language: | eng |
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