Lattices from Elliptic Curves over Finite Fields

In their well known book Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities

Lattice methods for finding rational points on varieties over number fields

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