This paper aims at studying the Iwasawa $λ$-invariant of the $p$-primary Selmer group. We study the growth behaviour of $p$-primary Selmer groups in $p$-power degree extensions over non-cyclotomic $\mathbb{Z}_p$-extensions of a number field. We prove a generalization of Kida's formula in such a case. Unlike the cyclotomic $\mathbb{Z}_p$-extension, where all primes are finitely decomposed; in the $\mathbb{Z}_p$-extensions we consider, primes may be infinitely decomposed. In the second part of the paper, we study the relationship for Iwasawa invariants with respect to congruences, obtaining refinements of the results of R. Greenberg-V. Vatsal and K. Kidwell. As an application, we provide an algorithm for constructing elliptic curves with large anticyclotomic $λ$-invariant. Our results are illustrated by explicit computation.
The Kida's formula in classical Iwasawa theory relates the Iwasawa $λ$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $λ$-invariants of Selmer groups under an appropriate $μ=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $μ\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.
We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes.
We prove a slight generalization of Iwasawa's `Riemann-Hurwitz' formula for number fields and use it to generalize Ferrero's and Kida's well-known computations of Iwasawa λ-invariants for the cyclotomic Z_2-extensions of imaginary quadratic number fields. In particular, we show that if p is a Fermat prime, then similar computations of Iwasawa λ-invariants hold for certain imaginary quadratic extensions of the unique subfield k of Q(ζ_{p^2}) such that [k:Q] = p. In fact, we actually prove more by explicitly computing cohomology groups of principal ideals. The computation of lambda invariants ob…
Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M.~Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on $λ$-invariants in a $p$-extension of $\mathbb{Z}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa's second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched $\mathbb{Z}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the…
We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weak…
We study the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions in arithmetic statistics and Iwasawa theory.
In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the Birch and Swinnerton-Dyer conjectures for analytic rank 0 or 1.
Bleher et al. began studying higher codimension Iwasawa theory for classical Iwasawa modules. Subsequently, Lei and Palvannan studied an analogue for elliptic curves with supersingular reduction. In this paper, we obtain a vast generalization of the work of Lei and Palvannan. A key technique is an approach to the work of Bleher et al. that the author previously proposed. For this purpose, we also study the structure of $\pm$-norm subgroups and duality properties of multiply-signed Selmer groups.
This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.
We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the Beilinson-Kato element to the equivariant $p$-adic $L$-functions. Then we propose equivariant main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application, we prove a case of a conjecture of Mazur-Tate.
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In p…
Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dy…
In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Z_p-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi and Perrin-Riou, we define restricted Selmer groups and λ^\pm, μ^\pm-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic Z_p-extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime …
We study this subject by first proving that the p-primary subgroup of the classical Selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the Galois module of p-power torsion points. We then prove theorems of Mazur, Schneider, and Perrin-Riou on the basis of this description. The final section, which is half of this long paper, contains a number of results and examples including a thorough study of the mu-invariant.
Let p be an odd prime. Suppose that E is a modular elliptic curve/Q with good ordinary reduction at p. Let Q_{oo} denote the cyclotomic Z_p-extension of Q. It is conjectured that Sel_E(Q_{oo}) is a cotorsion Lambda-module and that its characteristic ideal is related to the p-adic L-function associated to E. Under certain hypotheses we prove that the validity of these conjectures is preserved by congruences between the Fourier expansions of the associated modular forms.
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's $2$-variable $p$-adic $L$-functions) and algebraic objects (two "everywhere unramified'' Iwasawa modules) involving codimension two cycles in a $2$-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$, defined over $\…
We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension $Q(μ_p^\infty,\sqrt[p^\infty]{m})/Q$. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory.
We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$ with the good growth properties of the classical Pollack $p$-adic $L$-functions that in fact match them exactly when $a_p=0$ and $p$ is odd. We then generalize Kobayashi's methods to define two Selmer groups $\Sel^{\sharp}$ and $\Sel^{\flat}$ and formulate a main conjecture, stating that each characteristic ideal of the duals of these…
Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension to study the dual Selmer group over the $PGL(2)$ extension.
Computing the Iwasawa $μ$-Invariants for Elliptic Curves over $\mathbb{Z}_2$-Extensions
The University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Morningside Center of Mathematics（数学与系统科学研究院；中国科学院）；Academy of Mathematics and Systems Sciences, Chinese Academy of Science
We show that Riemann-Hurwitz-style translation formulas obtained by Kuz'min, Kida, Iwasawa, Wingberg et alii for the lambda invariant attached to certain Iwasawa moduli in cyclotomic Z{\ell}-extension of number fields are essentially equivalent. More precisely, we prove that all these formulas,including those stated in terms of representations, resul tidentically for purely algebraic reasons from the arithmetic computation of a suitable Herbrand quotient which it suffices to carry out in the cyclic case of prime degree {\ell}.
Kida, Rikuna and Sato [KRS10] developed a classification theory for Brumer's quintic polynomials via Kummer theory arising from associated elliptic curves. We generalize their results to elliptic curves associated to Lecacheux's quintic $F_{20}$-polynomials instead of Brumer's quintic $D_5$-polynomials.
We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bound…
Let $p >= 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\infty=Q(E[p^\infty])$ be a pro-$p$ Galois extension over a number field $K$. We consider $X(E/K_\infty)$, the Pontryagin dual of the $p$-Selmer group $\Sel_{p^\infty}(E/K_\infty)$. The size of this module is roughly measured by its rank $τ$ over a $p$-adic Galois group algebra $Λ(H)$, which has been studied in the past decade. We prove $τ>= 2$ for almost every elliptic curve under standard assumptions. Following from a result of Coates et al, $τ$ is odd if and only if $[Q(E[p]) \colon Q]/2$ is odd; we …
For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a fine signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.
We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.
Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension $\mathcal{F}_\infty$ of $\mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ of $E$ over $\mathcal{F}_\infty$. In this paper, we study the arithmetic statistics for the algebraic structure of this Selmer group. The results provide insights into the asymptotics for the growth of Mordell--Weil ranks of elliptic curves in noncommutative towers.
We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the indefinite cases) as well as the $\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman maps, which we use to introduce doubly-signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Fl…
We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension $D_\infty$ of the CM field $K$, where $p$ is a prime of good, supersingular reduction for $E$. Our main result yields an asymptotic formula for the corank of the $p$-primary Selmer group of $E$ along the extension $D_\infty/K$.
In our previous work, by using Kolyvagin derivatives of elliptic units, we constructed ideals C_i of the Iwasawa algebra, and proved that the ideals C_i become "upper bounds" of the higher Fitting ideals of the one and two variable p-adic unramified Iwasawa module X over an abelian extension field K_0 of an imaginary quadratic field K. In this article, by using "non-arithmetic" specialization arguments, we prove that the ideals C_i also become "lower bounds" of the higher Fitting ideals of X. In particular, we show that the ideals C_i determine the pseudo-isomorphism class of X. Note that in t…
This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (ϕ,Γ)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend i…
