This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curve.Expand

Polynomials over Finite Fields.- Primes, Arithmetic Functions, and the Zeta Function.- The Reciprocity Law.- Dirichlet L-series and Primes in an Arithmetic Progression.- Algebraic Function Fields and… Expand

Abstract. Nagao has recently given a conjectural limit formula for the rank of an elliptic surface E in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture on… Expand

Let D denote an integer congruent to 0 or l modulo 4 and not a square. Then D can be uniquely written äs D — D0m where Z)0, m e Z and either D0 is square-free and D0 = l (mod4) or D0 = 4m0 with m0… Expand

This chapter is devoted to the explanation and, in special cases, the proof of a conjecture which generalizes the famous theorem of Stickelberger about the structure of the class group of cyclotomic… Expand

In the ring Ok of algebraic integers of a number field K the group Ik of ideals of Ok modulo the subgroup Pk of principal ideals is a finite abelian group of order hk , the class number of K. The… Expand