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 .

We prove that for any infinite-type orientable surface S there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has infinite complement in C(S), the curve complex of S. As a consequence we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

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Let p be a prime, ε > 0 and 0 < L + 1 < L + N < p. We prove that if p 1/2+ε < N < p 1−ε , thenWe use this bound to show that any λ ≡ 0 (mod p) can be represented in the form λ ≡ n 1 ! . . . n 7 ! (mod p), where n i = o(p 11/12 ). This slightly refines the previously known range for n i .