論文

We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank.

In this paper, we formulate a new conjecture concerning Kato’s Euler system for elliptic curves E over Q. This ‘Generalized Perrin-Riou Conjecture’ predicts a precise congruence relation between a Darmon-type derivative of the zeta element of E over an arbitrary real abelian field and the critical value of an appropriate higher derivative of the L-function of E over Q. We prove the conjecture specializes in the relevant case of analytic rank one to recover Perrin-Riou’s conjecture on the logarithms of zeta elements, and also that, under mild technical hypotheses, the ‘order of vanishing’ part of the conjecture is unconditionally valid in arbitrary rank. This approach also allows us to prove a natural higher-rank generalization of Rubin’s formula concerning derivatives of p-adic L-functions and to establish an explicit connection between the p-part of the classical Birch and Swinnerton-Dyer formula and the Iwasawa main conjecture in arbitrary rank and for arbitrary reduction at p. In a companion article we prove that the approach developed here also provides a new interpretation of the Mazur–Tate conjecture that leads to the first (unconditional) theoretical evidence in support of this conjecture for curves of strictly positive rank.

We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories.

For a rational elliptic curve E and a positive integer n, satisfying some properties, we introduce analytic quantities delta_{n} using modular symbols, and give conjectures that these quantities control the maps of reduction modulo primes dividing n on E(Q). These conjectures also describe the structure of Selmer groups and the Tate-Shafarevich group of E. One of the aims of this paper is to provide an exposition on the theory of these analytic quantities. In the direction of the conjectures, we generalize, to all good reduction primes p, an injectivity theorem which was proven only for good ordinary primes in our earlier paper.

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic Zp-extension K∞/K, we consider XK∞,S=Gal(MK∞,S/K∞) where S is a finite set of places of k containing all ramifying places in K∞ and archimedean places, and MK∞,S is the maximal abelian pro-p-extension of K∞ unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of XK∞,S as a Zp[[Gal(K∞/k)]]-module, using the p-rank of Gal(K/k). This result explains the complexity of XK∞,S as a Zp[[Gal(K∞/k)]]-module when the p-rank of Gal(K/k) is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of XK∞,S, and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of XK∞,S.

Our aim in this paper is to give a clear view of what can be proved on the Fitting ideal of the Pontryagin dual of the minus class group of a CM-field as a Galois module. The class group we study is the classical full class group, and not the ray class group mod T (we are particularly interested in the Teichmüller character component of the class group). We show that several theorems in the existing literature hold unconditionally, using recent groundbreaking results by Dasgupta, Kakde, Silliman, and Wang. In particular, we give a simple proof of the equivariant Iwasawa main conjecture including the case p=2, using their keystone theorems due to Dasgupta and Kakde in [Ann. of Math. (2) 197 (2023), 289–388] and due to Dasgupta, Kakde, Silliman and Wang in [arXiv:2310.16399]. We unconditionally compute the Fitting ideal of the Pontryagin dual of the minus class group of the cyclotomic Zp-extension of a CM-field, and also the Fitting ideal of S-ramified Iwasawa modules for totally real number fields. The numbers of minimal generators of these Iwasawa modules are also studied.

Dans cet article, pour une extension abélienne K/k de corps de nombres de type CM, nous proposons une conjecture qui décrit complètement l’idéal de Fitting de la partie moins du dual de Pontryagin du groupe de classes de rayon T de K, pour un ensemble T d’idéaux premiers, comme Gal(K/k)-module. Nous soulignons que nous considérons ici le groupe de classes au sens propre, sans laisser de côté les idéaux ramifiés (l’objet que nous étudions n’est pas le quotient du groupe de classes par le sous-groupe engendré par les classes des idéaux premiers ramifiés). Nous prouvons que notre conjecture est une conséquence de la conjecture de nombres de Tamagawa équivariante, et prouvons la version de notre conjecture en théorie d’Iwasawa.

In this article we shall generalize the theory of Stark elements to arbitrary ‘weight’. This means that we define canonical generalized Stark elements that correspond to the leading terms at arbitrary integers of L-series attached to the multiplicative group. We investigate their basic properties, and formulate both a conjecture on the precise integrality condition that these Stark elements satisfy and a conjecture that describes the Galois module structures of certain cohomology groups as a generalization of the refined Rubin-Stark conjecture we formulated in an earlier article ‘On zeta elements for Gm’ in Doc. Math. 21 (2016). We then prove these conjectures in several interesting cases. The integrality condition studied here will then play an important role in a subsequent article of the first and third authors which formulates a precise conjectural congruence relation between the generalized Stark elements of differing weights that are defined here.

We develop an explicit `higher rank’Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of number fields. We show this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture and, as a first application of this approach, we prove new cases of the conjecture over natural families of abelian CM-extensions of totally real fields for which the relevant p-adic L-functions possess trivial zeroes.

For a finite abelian p-extension K/k of totally real fields and the cyclotomic Zp-extension K∞/K, we prove a strong version of an equivariant Iwasawa main conjecture by determining completely the Fitting ideal over Zp[[Gal(K∞/k)]] of the classical Iwasawa module X_{K∞,S}, which is the Galois group of the maximal pro-p abelian S-ramified extension of K∞, where S contains all ramified primes in K∞/k. To do this, we prove a conjecture which was proposed in a previous paper by the first and second author, concerning the minors of a relation matrix for the second syzygy module of the trivial module Z over a suitable group ring.

In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main conjecture"à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture"suggested by the second named author under the validity of the $\pm $-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm $-objects", and "fine objects"in Iwasawa theory. The case of good ordinary reduction is also treated.