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Title: Lattice methods for finding rational points on varieties over number fields
Author: Turner, Charlotte L.
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2013
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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.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics