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.

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

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 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.

In this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple $\mathbb{Z}_{p}$-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the $\unicode[STIX]{x1D707}=0$ conjecture for cyclotomic $\mathbb{Z}_{p}$-extensions. We show that in certain non-cyclotomic $\mathbb{Z}_{p}$-towers, the $\unicode[STIX]{x1D707}$-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified $p$-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-$p$-adic analytic.

We study the variation of $\unicode[STIX]{x1D707}$-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$-adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$-adic $L$-function is simply a power of $p$ up to a unit (i.e. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Let $p$ be a prime and let $G$ be a finite group. By a celebrated theorem of Swan, two finitely generated projective $\mathbb{Z}_{p}[G]$-modules $P$ and $P^{\prime }$ are isomorphic if and only if $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$ and $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$ are isomorphic as $\mathbb{Q}_{p}[G]$-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.

We prove an interpolation formula for the values of certain $p$-adic Rankin-Selberg $L$-functions associated with non-ordinary modular forms.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$-extension.

For the $(d+1)$-dimensional Lie group $G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$, we determine through the use of $p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian $L$-functions arises from an element in $K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$. If $E$ is a semistable elliptic curve over $\mathbb{Q}$, these abelian $L$-functions already exist; therefore, one can obtain many new families of higher order $p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

Let $E$ be an elliptic curve over $\mathbb{Q}$ that has good supersingular reduction at $p\,>\,3$. We construct what we call the $\pm /\pm $-Selmer groups of $E$ over the $\mathbb{Z}_{p}^{2}$-extension of an imaginary quadratic field $K$ when the prime $p$ splits completely over $K/\mathbb{Q}$, and prove that they enjoy a property analogous to Mazur's control theorem.

Furthermore, we propose a conjectural connection between the $\pm /\pm $-Selmer groups and Loeffler's two-variable $\pm /\pm $-$p$-adic $L$-functions of elliptic curves.

For an odd prime p we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over F(μp∞), the field obtained from a totally real number field F by adjoining all p-power roots of unity. We use a new approach based on the relationship between eigenspaces and étale cohomology groups over the cyclotomic ℤp-extension F∞ of F. The systematic use of étale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces.

Suppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i (){p}.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

Let p be an odd prime and L/F a p-extension of number fields with Galois group G. The aim of this paper is to provide answers to a question of Kahn concerning lower bounds for the order of the kernel and cokernel of the functorial map K2F → K2LG. To this end, we first determine a norm index formula for generalized Tate kernels and then express our lower bounds in terms of the ramification in L/F.