We give a short proof of the anticyclotomic analogue of the “strong” main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic $\mathbb {Z}_p$-extension in terms of Bertolini–Darmon’s theta elements.

Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.

Let p be an odd prime, and suppose that $E_1$ and $E_2$ are two elliptic curves which are congruent modulo p. Fix an Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\mathbb{C})$ over a totally real field F, induced from a Hecke character over a CM-extension $K/F$. Assuming $E_1$ and $E_2$ are ordinary at p, we compute the variation in the $\mu$- and $\lambda$-invariants for the $\tau$-part of the Iwasawa Main Conjecture, as one switches from $E_1$ to $E_2$. Provided an Euler system exists, it will follow directly that IMC$(E_1,\tau)$ is true if and only if IMC$(E_2,\tau)$ is true.

This paper discusses variants of Weber’s class number problem in the spirit of arithmetic topology to connect the results of Sinnott–Kisilevsky and Kionke. Let p be a prime number. We first prove the p-adic convergence of class numbers in a ${\mathbb{Z}_{p}}$-extension of a global field and a similar result in a ${\mathbb{Z}_{p}}$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the p-adic limit of the p-power-th cyclic resultants of a polynomial using roots of unity of orders prime to p, the p-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with p-adic limits being in ${\mathbb{Z}}$ and find the cases such that the base p-class numbers are small and $\nu$’s are arbitrarily large.

Let $K={\mathbb {Q}}(\sqrt {-7})$ and $\mathcal {O}$ the ring of integers in $K$. The prime $2$ splits in $K$, say $2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$. Let $A$ be an elliptic curve defined over $K$ with complex multiplication by $\mathcal {O}$. Assume that $A$ has good ordinary reduction at both $\mathfrak {p}$ and ${\mathfrak {p}}^*$. Write $K_\infty$ for the field generated by the $2^\infty$–division points of $A$ over $K$ and let ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$. In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable $2$-adic $L$-function on $\mathcal {G}$. Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa $\mu$ invariant zero.

Let $E/\mathbb {Q}$ be an elliptic curve and let p be a prime of good supersingular reduction. Attached to E are pairs of Iwasawa invariants $\mu _p^\pm $ and $\lambda _p^\pm $ which encode arithmetic properties of E along the cyclotomic $\mathbb {Z}_p$-extension of $\mathbb {Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $\mu _p^\pm =0$. We provide support for this conjecture by proving that for any $\ell \geq 0$, we have $\mu _p^\pm \leq 1$ for all but finitely many primes p with $\lambda _p^\pm =\ell $. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $\mu _p^\pm \leq 1$ holds on a density 1 set of good supersingular primes for E.

Let E be an elliptic curve defined over $\mathbb {Q}$ with good ordinary reduction at a prime $p\geq 5$ and let F be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell–Weil group of E over the $\mathbb {Z}_{p}^2$-extension of F is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb {Z}_{p}^2$.

Let p be a fixed prime number, and let F be a global function field with characteristic not equal to p. In this article, we shall study the variation properties of the Sylow p-subgroups of the even K-groups in a p-adic Lie extension of F. When the p-adic Lie extension is assumed to contain the cyclotomic $\mathbb {Z}_p$-extension of F, we obtain growth estimate of these groups. We also establish a duality between the direct limit and inverse limit of the even K-groups.

In this article, we obtain a necessary and sufficient condition for the pseudo-nullity of the p-ramified Iwasawa module for p-adic Lie extensions of totally real fields. It is applied to answer the corresponding question for the minus component of the unramified Iwasawa module for CM-fields. The results show that the pseudo-nullity is very rare.

In this article, we generalize results of Clozel and Ray (for $SL_2$ and $SL_n$, respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-p Iwahori subgroup of a simple, simply connected, split group $\mathbf {G}$ over ${{\mathbb Q}_p}$.

We prove a comparison theorem between Greenberg–Benois $\mathcal {L}$-invariants and Fontaine–Mazur $\mathcal {L}$-invariants. Such a comparison theorem supplies an affirmative answer to a speculation of Besser–de Shalit.

In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

Let p be a prime. In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of degree p. We also study growth of the p-primary Selmer group and the Shafarevich–Tate group in cyclic degree-p extensions and improve upon previously known results in this direction.

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.

We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$-extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty .$ For example, for each tower in a basic class of ${\mathbf Z}_p$-towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$-extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$. We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.