Lattice methods for finding rational points on varieties over number fields

Doctoral thesis English OPEN
Turner, Charlotte L.
  • Subject: QA

We develop a method for finding all rational points of bounded height on a variety\ud defined over a number field K. Given a projective variety V we find a prime p\ud of good reduction for V with certain properties and find all points on the reduced\ud curve V (Fp). For each point P 2 V (Fp) we may define lattices of lifts of P: these\ud lattices contain all points which are congruent to P mod p satisfying the defining\ud polynomials of V modulo a power of p. Short vectors in these lattices are possible\ud representatives for points of bounded height on the original variety V (K). We make\ud explicit the relationship between the length of a vector and the height of a point\ud in this setting. We will discuss methods for finding points in these lattices and\ud how they may be used to find points of V (K), including a method involving lattice\ud reduction over number fields.\ud The method is implemented in Sage and examples are included in this thesis.
  • References (1)

    96 [23] Huguette Napias. A generalization of the LLL-algorithm over euclidean rings or orders. Journal de Theorie des Nombres de Bordeaux, 8:387{396, 1996. [24] Jurgen Neukirch. Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften). Springer, 1999.

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