Euler systems are certain norm-compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou predicting what kind of Euler system one should expect for a general Galois representation. Surprisingly, several recent constructions of Euler systems do not seem to fit the predictions of this conjecture, and we formulate a more general conjecture which explains these extra objects. The novel aspect of our conjecture is that it predicts that there should often be Euler systems of several different ranks associated to a given Galois representation, and we describe how we expect these objects to be related.

To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the $R$-annihilator $\mathrm{Ann}_{R}(M)$ of $M$, but is often much easier to compute. This notion has recently been generalised to that of so-called ‘Fitting invariants’ over certain noncommutative rings; the present author considered the case in which $R$ is an $\mathfrak{o}$-order $\Lambda$ in a finite dimensional separable algebra, where $\mathfrak{o}$ is an integrally closed commutative noetherian complete local domain. This article is a survey of known results and open problems in this context. In particular, we investigate the behaviour of Fitting invariants under direct sums. In the appendix, we present a new approach to Fitting invariants via Morita equivalence.

In this paper we give an overview of some aspects of Iwasawa theory for modular forms. We start with the classical formulation in terms of $p$-adic $L$-functions in the ordinary case and the $\pm$-formulation for supersingular elliptic curves. Then we discuss some recent progresses in the proof of the corresponding Iwasawa main conjectures formulated by Kato (Conjecture 4.1), which relates the index of his zeta element to the characteristic ideal of the strict Selmer groups.

Let $L/k$ be a finite abelian extension of an imaginary quadratic number field $k$. Let $\mathfrak{p}$ denote a prime ideal of $\mathcal{O}_k$ lying over the rational prime $p$. We assume that $\mathfrak{p}$ splits completely in $L/k$ and that $p$ does not divide the class number of $k$. If $p$ is split in $k/\mathbb{Q}$ the first named author has adapted a construction of Solomon to obtain elliptic $\mathfrak{p}$-units in $L$. In this paper we generalize this construction to the non-split case and obtain in this way a pair of elliptic $\mathfrak{p}$-units depending on a choice of generators of a certain Iwasawa algebra (which here is of rank 2). In our main result we express the $\mathfrak{p}$-adic valuations of these $\mathfrak{p}$-units in terms of the $p$-adic logarithm of an explicit elliptic unit. The crucial input for the proof of our main result is the computation of the constant term of a suitable Coleman power series, where we rely on recent work of T. Seiriki.

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 $\mathbb{G}_m$’ 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.

Our first goal in this article is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson–Kato classes follows as an easy consequence of the Iwasawa main conjectures. We also explain that the refined form of this conjecture in the $p$-supersingular case also follows from the classical Gross–Zagier formula and Kobayashi's $p$-adic Gross–Zagier formula combined with this simple observation.

Our second goal is to set up a conceptual framework in the context of $\Lambda$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of CM abelian varieties and the (conjectural) Coleman–Rubin–Stark elements we introduce here. This can ben thought of as a higher dimensional version of Rubin's results on rational points on CM elliptic curves.

In this note, we revisit Hida's construction of $p$-adic Rankin-Selberg $L$-functions by incorporating Jacquet's approach to automorphic $L$-functions on $\mathrm{GL}(2) \times \mathrm{GL}(2)$. This allows us to give a construction of primitive three variable $p$-adic Rankin-Selberg $L$-functions associated with a pair of two primitive Hida families in full generality and prove the functional equation of this $p$-adic Rankin-Selberg $L$-function.

In this note, we provide a new proof of the rank 1 Gross–Stark conjecture for a quadratic extension under the assumption that there is only one prime in the base field above a rational prime $p$. The full Gross–Stark conjecture was proven by the authors in joint work with Ventullo, building on a prior result in the rank 1 setting by the first named author in joint work with Darmon and Pollack. The proof given in this note is much simpler as it does not use the theory of $p$-adic Galois cohomology and Galois representations associated to $p$-adic modular forms. Instead, the proof relies on a certain explicit construction using Theta series, congruences with Eisenstein series and the $q$-expansion principle of Deligne–Ribet.

We introduce a natural way to define Selmer groups and $p$-adic $L$-functions for modular forms of weight 1. The corresponding Galois representation $\rho$ of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is a 2-dimensional Artin representation with odd determinant. Thus, the dimension $d^{+}$ of the (+1)-eigenspace for complex conjugation is 1. Choose a prime $p$ such that the restriction of $\rho$ to the local Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ has a 1-dimensional constituent $\varepsilon$ with multiplicity 1. If we fix the choice of such an $\varepsilon$, we can define a Selmer group and a $p$-adic $L$-function. On the algebraic side, we prove that the Selmer group over the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$ is a cotorsion module over the Iwasawa algebra $\Lambda$. That result is valid for an Artin representation of arbitrary dimension $d$ under the assumption that $d^{+} = 1$ and that such an $\varepsilon$ can be chosen. On the analytic side, the corresponding complex $L$-function has no critical values and the definition of the $p$-adic $L$-function depends on deforming the Galois representation $\rho$ by Hida theory.

In this note we propose an analog of the well-known Cohen-Lenstra heuristics for modules over the Iwasawa algebra $\Lambda$. It turns out that only the analog of the real-quadratic situation leads to a convergent series and hence to potential predictions. We determine the sum of this series, which runs over all isomorphism classes of finite $\Lambda$-modules, and we discuss the partial sum that arises by restricting to cyclic $\Lambda$-modules. We demonstrate that this subsum is almost as large as the total sum. No attempt is made to test the heuristics numerically.

For a finite abelian $p$-extension $K/k$ of totally real fields and the cyclotomic $\mathbb{Z}_{p}$-extension $K_{\infty}/K$, we prove a strong version of an equivariant Iwasawa main conjecture by determining completely the Fitting ideal over $\mathbb{Z}_{p}[[\mathop{\rm Gal}\nolimits(K_{\infty}/k)]]$ of the classical Iwasawa module $X_{K_{\infty},S}$, which is the Galois group of the maximal pro-$p$ abelian $S$-ramified extension of $K_{\infty}$, where $S$ contains all ramified primes in $K_{\infty}/k$. 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 $\mathbb{Z}$ over a suitable group ring.

We study the universal minimal ordinary Galois deformation $\rho_{\mathbb{T}}$ of an induced representation $\operatorname{Ind}_F^{\mathbb{Q}} \varphi$ from a real quadratic field $F$ with values in $\mathrm{GL}_2(\mathbb{T})$. By Taylor–Wiles, the universal ring $\mathbb{T}$ is isomorphic to a local ring of a Hecke algebra. Combining an idea of Cho–Vatsal [CV03] with a modified Taylor–Wiles patching argument in [H17], under mild assumptions, we show that the Pontryagin dual of the adjoint Selmer group of $\rho_{\mathbb{T}}$ is canonically isomorphic to $\mathbb{T}/(L)$ for a non-zero divisor $L \in \mathbb{T}$ which is a generator of the different $\mathfrak{d}_{\mathbb{T}/\Lambda}$ of $\mathbb{T}$ over the weight Iwasawa algebra $\Lambda=W[[T]]$ inside $\mathbb{T}$. Moreover, defining $\langle\varepsilon\rangle := (1+T)^{\log_p(\varepsilon)/\log_p(1 + p)}$ for a fundamental unit $\varepsilon$ of the real quadratic field $F$, we show that the adjoint Selmer group of $\operatorname{Ind}_F^\mathbb{Q}\Phi$ for the (minimal) universal character $\Phi$ deforming $\varphi$ is isomorphic to $\Lambda/(\langle\varepsilon\rangle - 1)$ as $\Lambda$-modules.

The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial) Fitting ideals of the Iwasawa modules for the cyclotomic $\mathbb{Z}_p$-extensions of finite abelian extensions of totally real fields. In this paper, we generalize the algebraic theory behind their work by developing the theory of “shifts of Fitting invariants.” As applications to Iwasawa theory, we obtain a noncommutative version and a two-variable version of the work of Greither and Kurihara.

This is the Part II of our paper “Height functions for motives”. We consider more general period domains and the height functions on more general sets of motives. We also consider the corresponding Hodge theoretic variant of Nevanlinna theory.

We prove a one-sided divisibility relation for the anticyclotomic Iwasawa main conjecture for modular forms in terms of the $p$-adic $L$-function constructed by Bertolini-Darmon-Prasanna and Brakočević. The divisibility relation is known by Castella if $p$ is ordinary and by Castella-Wan for the elliptic curve case. Here we prove the higher weight non-ordinary case with a treatment that works uniformly for both ordinary and non-ordinary cases. In the proof, we establish a theory of integral Perrin-Riou twist. It enables us not only to twist systems of generalized Heegner cycles (which are not norm-compatible) by any continuous $p$-adic anticyclotomic characters but also to investigate the denominators of resulting systems explicitly.

In this paper, we derive a formula for the $p$-adic syntomic regulators of Asai–Flach classes. These are cohomology classes forming an Euler system associated to a Hilbert modular form over a quadratic field, introduced in an earlier paper [LLZ16] by Antonio Lei and the first and third authors. The formula we develop here is expressed in terms of differential operators acting on overconvergent Hilbert modular forms; it is analogous to existing formulae for the regulators of Beilinson–Flach classes, but a novel feature is the appearance of a projection operator associated to a critical-slope Eisenstein series. We conclude the paper with numerical calculations giving strong evidence for the non-vanishing of these regulators in an explicit example.

For an odd prime number $p$ and a number field $k$ which is an elementary abelian $p$-extension of the rationals, we prove the equivalence between the vanishing of all Iwasawa invariants of the cyclotomic $\mathbb{Z}_p$-extension of $k$ and an arithmetical condition described by the linking numbers of primes from a viewpoint of analogies between pro-$p$ Galois groups and link groups. A criterion of Greenberg's conjecture for $k$ of degree $p$ is also described in terms of linking matrices.

Jannsen established several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of $p$-adic Lie groups) over $\mathbb{Z}_p$, which are very useful for determining certain properties of such modules in arithmetic applications. Slight generalizations of said results are also obtained by Nekovář (for abelian groups and more general coefficient rings), by Venjakob (for products of not necessarily abelian groups, but with $\mathbb{Z}_p$-coefficients), and by Lim-Sharifi. Unfortunately, some of Jannsen's spectral sequences for families of representations as coefficients for (local) Iwasawa cohomology are still missing. We explain and follow the philosophy that all these spectral sequences are consequences or analogues of local cohomology and duality à la Grothendieck (and Tate for duality groups).