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The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1. In the 1960’s the pione...

Let E be an optimal elliptic curve of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of...

Let E be an optimal elliptic curve over Q of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of the sub...

It is a standard remark that the group Aut(C) of abstract field automorphisms of C has uncountably many elements but only two that are are continuous. One consequence is that the analytic properties of a Dirichlet series ∑ n>1 a(n)n −s do not usually carry over to ∑ n>1 ι(a(n))n −s for ι ∈ Aut(C). Silly counterexamples abound: for instance, take a(n) = (1 − √ 2) and choose ι so that ι( √ 2) = −...

CALE M. GENTRY,* JEFFREY M. SHAINLINE, MARK T. WADE, MARTIN J. STEVENS, SHELLEE D. DYER, XIAOGE ZENG, FABIO PAVANELLO, THOMAS GERRITS, SAE WOO NAM, RICHARD P. MIRIN, AND MILOŠ A. POPOVIĆ Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA Depart...