Abstract. We introduce several new methods to obtain upper bounds on the number of solutions of the congruenceswith a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curvesover the finite field F p of p elements, with coefficients in a 2g-dimensional cubethat are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over F p .

Abstract. We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets, B h [g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "A contains a non-trivial solution of M ·x = 0" where A is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1, . . . , n}. Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation.Furthermore, we study the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.

Given a sequence of distinct positive integers v 1

For a prime p and a given square box, B, we consider all elliptic curves E r,s : Y 2 = X 3 + rX + s defined over a field F p of p elements with coefficients (r, s) ∈ B. We obtain a nontrivial upper bound for the number of such curves which are isomorphic to a given one over F p , in terms of the size of B. We also give an optimal lower bound on the number of distinct isomorphic classes represented.

Abstract. We study the typical behavior of the least common multiple of the elements of a random subset A ⊂ {1, . . . , n}. For example we prove that lcm{a : a ∈ A} = 2 n(1+o(1)) for almost all subsets A ⊂ {1, . . . , n}.