We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g = 1. We also show when g = 1 or g = 2 that our bounds are best possible. 1

Abstract not found

Abstract. We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields Fqm where either q is fixed or m = 1 and q is prime. Here we let both q and m vary and our estimate is explicit and does not depend on the elliptic curve. 1. introduction Let Fq be a finite fields with q = p m elements and let E be an elliptic curve defined over Fq. It is well known (see for example the book of Washington [7]) that the group of rational point of E over Fq satisfies E(Fq) ∼ = Zn × Znk where n, k ∈ N are such that n | q − 1. The exponent of E(Fq) is exp(E(Fq)) = nk. In 1989 Schoof [6] proved that if E is an elliptic curve over Q without complex multiplication, then for every prime p> 2 of good reduction for E, one has the estimate √ log p exp(E(Fp))> CE p (log log p) 2 where CE> 0 is a constant depending only on E. In 2005 Luca and Shparlinski [4] consider the case when q is fixed and they prove that if E/Fq is ordinary, the there exists an effectively computable constant ϑ(q) depending only on q such that log m (1) exp(E(Fqm))> qm/2+ϑ(q)m/ holds for all positive integers m. Other lower bounds that hold for families of primes (resp. for families of powers of fixed primes) with density one were proven by Duke in [1] (resp. by Luca and Shparlinski in [4]). Here we let both p and m vary and we prove the following Theorem. Let E be any elliptic curve over Fpm is even and E(F p 2r) ∼ = Zp r ±1 × Zp r ±1 where m ≥ 3 then either m = 2r

A lower bound for the r-order of a matrix modulo N

In this paper, we prove that the period of the continued fraction expansion of √ 2 n + 1 tends to infinity when n tends to infinity through odd positive integers. 1