Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.

/pdf/on-the-birch-swinnerton-dyer-quotients-modulo-squares-1q9ugl3rr8.pdf

On the Birch-Swinnerton-Dyer quotients modulo squares

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K i , s) for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of LII, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of III(E/K(E[2]))[6 ∞ ] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p ∞ -Selmer rank and the analytic rank agree.

http://annals.math.princeton.edu/wp-content/uploads/annals-v172-n1-p11-p.pdf

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p
 ∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K
 ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of $\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})$
 
. We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

Regulator constants and the parity conjecture

This is an introduction to Iwasawa theory and its generalizations. We discuss some
main conjectures and related subjects.

http://icm2006.org/proceedings/Vol_I/18.pdf

Iwasawa theory and generalizations

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z
p. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

/pdf/the-gl-2-main-conjecture-for-elliptic-curves-without-complex-2fu85x1c2m.pdf

The GL 2 Main Conjecture for Elliptic Curves without Complex Multiplication

The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex L-functions, typified by the conjecture of Birch and Swinnerton-Dyer and its generalizations. Our goal in the present paper is to develop algebraic techniques which enable us to formulate a precise version of such a main conjecture for motives over a large class of p-adic Lie extensions of number fields. The paper ends by formulating and briefly discussing the main conjecture for an elliptic curve E over the rationals Q over the field generated by the coordinates of its p-power division points, where p is a prime greater than 3 of good ordinary reduction for E.

The GL_2 main conjecture for elliptic curves without complex multiplication

Let E be an elliptic curve over a number field F, and let F ∞ be a Galois extension of F whose Galois group G is a p-adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E over F ∞ , and the global root numbers attached to the twists of the complex L-function of E by Artin representations of G.

/pdf/root-numbers-selmer-groups-and-non-commutative-iwasawa-2w6x64dd2q.pdf

Takako Fukaya

Papers

The GL 2 Main Conjecture for Elliptic Curves without Complex Multiplication

The GL_2 main conjecture for elliptic curves without complex multiplication

Root numbers, Selmer groups, and non-commutative Iwasawa theory