Abstract.In this expository paper we show how one can, in a uniform way, calculate the weight distributions of some well-known binary cyclic codes. The codes are related to certain families of curves, and the weight distributions are related to the distribution of the number of rational points on the curves.

Abstract. For every conductor f ∈ {1, 3, 4, 5, 7, 8, 9, 11, 12, 15} there exist non-zero abelian varieties over the cyclotomic field Q(ζ f ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X 1 (f ) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over Q(ζ f ) with good reduction everywhere except possibly when f = 11 or 15. Assuming the Generalized Riemann Hypothesis (GRH) we prove the same result when f = 11 and 15.

book reviews 111 2. M. Rieffel, Non-commutative tori-a case study of noncommutative differentiable manifolds,