For primes $$q \equiv 7 \ \mathrm {mod}\ 16$$ , the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field $$K = {\mathbb {Q}}(\sqrt{-q})$$ , which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.

Tate was born on March 13, 1925, and died on Oc

The paper uses Iwasawa theory at the prime p=2 to prove non‐vanishing theorems for the value at s=1 of the complex L ‐series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field K=Q(−q) , where q is any prime ≡7mod8 . Our results establish some broad generalizations of the non‐vanishing theorem first proven by Rohrlich using complex analytic methods. Such non‐vanishing theorems are important because it is known that they imply the finiteness of the Mordell–Weil group and the Tate–Shafarevich group of the corresponding elliptic curves over the Hilbert class field of K . It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to K of the particular elliptic curve with complex multiplication introduced by Gross.

In the late Victorian and Edwardian periods, several writers voiced their apprehensions about the state of the British Empire and the dangers they thought it faced by making comparisons between Britain and the Phoenician city of Tyre and the greatest of Tyre's colonies, Carthage. This paper compares Rider Haggard's use of this analogy in his novel Elissa or the Doom of Zimbabwe with other writers of his time who compared Britain to the Phoenicians. Haggard emerges as deeper, more wide-ranging and sophisticated in his use of the 'Phoenician analogy' than other writers who employed it.

The exact formulae of arithmetic geometry, most of which are still largely conjectural, are unquestionably some of the most mysterious, beautiful, and important questions in mathematics today. In these three lectures, I want to discuss the history of these exact formulae and the work which has been done on them in the past. Specifically, I intend to discuss the history of some key discoveries related to the following questions:(i) The procedure of infinite descent, (ii) Dirichlet’s class number formula, (iii) Kummer’s work on cyclotomic fields and Iwasawa’s profound extension of it, (iv) the conjecture of Birch and Tate, and (v) the conjecture of Birch and Swinnerton-Dyer. I will concentrate on the first key examples, rather than discussing the more general theorems which followed.

work has greatly influenced the study of diophantine geometry in the twentieth century, died at his home near Cambridge on December 26, 2018, at the age of ninety-one.He attended Eton College, where the photograph at right was taken.There he became interested in diophantine equations from reading Heath's translation of Diophantus of Alexandria, and wrote his first paper [21] on the equation 4 + 4 = 4 + 4 while still at Eton.Immediately after Eton, he went to Trinity College, Cambridge, and then, apart from a few years as a civil servant in London, he spent the rest of his life in Cambridge.In 1973 he became master of St. Catharine's College, Cambridge, and was vice-chancellor of the University of Cambridge from 1979 to 1981.

We propose a definition of thep-adic L-function of a motiveM over ℚ assumingM admits at least one critical point, andp is ordinary forM. This corrects by a power of\(i = \sqrt { - 1} \) an earlier definition of B. Perrin-Riou and the author.

(typeset by Robert JS McDonald, UConn1) Lecture 1. Foundational material. The lecture will briefly cover, without proofs, the background in algebra and number theory needed at the beginning of Iwasawa theory. Throughout, p will denote an arbitrary prime number, and Γ a topological group which is isomorphic to the additive group of p-adic integers Zp. Thus, for each n ≥ 0, Γ will have a closed subgroup of index pn, which we will denote by Γn, and Γ/Γn will then be a cyclic group of order pn. The Iwasawa algebra Λ(Γ) of Γ is defined by

The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X 0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X 0(49) by the quadratic extension \(KK(\sqrt M )/K\), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F ∞ = K(E p∞), where E p∞ denotes the group of p∞-division points on E. Moreover, writing B for the twist of X 0(49) by \(K(\sqrt[4]{{ - 7}})/K\), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.

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There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.