Iwasawa Algebra Research Articles

Skew power series rings over a prime base ring

In this paper, we investigate the structure of skew power series rings of the form S=R[[x;σ,δ]], where R is a complete, positively filtered ring and (σ,δ) is a skew derivation respecting the filtration. Our main focus is on the case in which σδ=δσ, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if P is an invariant prime ideal of R, is PS a prime ideal of S? When R has characteristic p, our results reduce this to a finite-index problem. We also give preliminary results in the “Iwasawa algebra” case δ=σ−idR in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is “almost” (σ,δ)-invariant in a certain sense.

On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting

Abstract We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.

A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

Abstract Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field K such that p is inert in K and that E has good reduction at p. Let $K_\infty$ be the anticyclotomic $\mathbb{Z}_p$-extension of K. Agboola–Howard defined Kobayashi-type signed Selmer groups of E over $K_\infty$ and showed that exactly one of them is cotorsion over the corresponding Iwasawa algebra. In this short note, we discuss a link between the characteristic ideals of the cotorsion signed Selmer group and the fine Selmer group building on a recent breakthrough of Burungale–Kobayashi–Ota on the structure of local points.

Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.

On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple ℤ p -Extensions Over Imaginary Quadratic Fields

For an odd prime number p, we study the number of generators of the unramified Iwasawa modules of the maximal multiple ℤ p -extensions over the Iwasawa algebra. In our previous paper, under several assumptions for an imaginary quadratic field, we obtained a necessary and sufficient condition for the cyclicity of the Iwasawa module over the Iwasawa algebra. The present work provides computational methods and numerical examples of Iwasawa modules that are cyclic as modules over the Iwasawa algebra. We remark that our methods do not require the assumption that Greenberg’s generalized conjecture holds.

Iwasawa Theory of Jacobians of Graphs

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff's Matrix Tree Theorem). A specific type of covering graph, called a derived graph, that is constructed from a voltage graph with voltage group $G$ is the object of interest in this paper. Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow $p$-subgroups of Jacobians in an infinite voltage $p$-tower, for any prime $p$, are obtained in terms of classical $\mu$ and $\lambda$ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.

Faithfulness of generalised Verma modules for Iwasawa algebras

We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type A2. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.

Fine Selmer groups and ideal class groups

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

On Greenberg's generalized conjecture

For a number field F and an odd prime number p, let F˜ be the compositum of all Zp-extensions of F and Λ˜ the associated Iwasawa algebra. Let GS(F˜) be the Galois group over F˜ of the maximal extension which is unramified outside p-adic and infinite places. In this paper we study the Λ˜-module XS(−i)(F˜):=H1(GS(F˜),Zp(−i)) and its relationship with X(F˜(μp))(i−1)Δ, the Δ:=Gal(F˜(μp)/F˜)-invariant of the Galois group over F˜(μp) of the maximal abelian unramified pro-p-extension of F˜(μp). More precisely, we show that under a decomposition condition, the pseudo-nullity of the Λ˜-module X(F˜(μp))(i−1)Δ is implied by the existence of a Zpd-extension L with XS(−i)(L):=H1(GS(L),Zp(−i)) being without torsion over the Iwasawa algebra associated to L, and which contains a Zp-extension F∞ satisfying H2(GS(F∞),Qp/Zp(i))=0. As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer i≡1mod[F(μp):F]. This existence is fulfilled for (p,i)-regular fields.

Dimension Theory in Iterated Local Skew Power Series Rings

Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull and global dimensions, obtaining lower bounds to complement the upper bounds obtained by Wang. In fact, we show that many common such rings obey a stronger property, which we call triangularity, and which allows us also to calculate their classical Krull dimension (prime length). Finally, we correct an error in the literature regarding the associated graded rings of general iterated skew power series rings, but show that triangularity is enough to recover this result.

SELMER GROUPS OF ELLIPTIC CURVES OVER THE EXTENSION

AbstractIwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

Primitive ideals in rational, nilpotent Iwasawa algebras

Given a p-adic field K and a nilpotent uniform pro-p group G, we prove that all primitive ideals in the K-rational Iwasawa algebra KG are maximal, and can be reduced to a particular standard form. Setting L as the associated Zp-Lie algebra of G, our approach is to study the action of KG on a Dixmier module D(λ)ˆ over the affinoid envelope U(L)ˆK, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.

Notes on the module of Euler systems for $p$-adic representations

We study the module of Euler systems for $p$-adic representations. We determine the ideal of an Iwasawa algebra associated with Euler systems of rank $0$. We also show that the module of higher rank Euler systems for $\mathbb G_{\mathrm m}$ over a totally

Fine Selmer groups of congruent p-adic Galois representations

AbstractWe compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions $K_\infty $ of number fields K. We prove that in several natural settings the $\pi $ -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu $ -invariants. In the special case of a $\mathbb {Z}_p$ -extension $K_\infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.

Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

P-adic Measures for Reciprocals of L-functions of Totally Real Number Fields

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

On Sharifi’s conjecture: Exceptional case

In the present article, we study the conjecture of Sharifi on the surjectivity of the map ϖ θ \varpi _{\theta } . Here θ \theta is a primitive even Dirichlet character of conductor N p Np , which is exceptional in the sense of Ohta. After localizing at the prime ideal p \mathfrak {p} of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt p p -adic L L -function L p ( s , θ − 1 ω 2 ) L_p(s,\theta ^{-1}\omega ^2) , we compute the image of ϖ θ , p \varpi _{\theta ,\mathfrak {p}} in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.

On fine Selmer groups and the greatest common divisor of signed and chromatic 𝑝-adic 𝐿-functions

Let E / Q E/\mathbb {Q} be an elliptic curve and p p an odd prime where E E has good supersingular reduction. Let F 1 F_1 denote the characteristic power series of the Pontryagin dual of the fine Selmer group of E E over the cyclotomic Z p \mathbb {Z}_p -extension of Q \mathbb {Q} and let F 2 F_2 denote the greatest common divisor of Pollack’s plus and minus p p -adic L L -functions or Sprung’s sharp and flat p p -adic L L -functions attached to E E , depending on whether a p ( E ) = 0 a_p(E)=0 or a p ( E ) ≠ 0 a_p(E)\ne 0 . We study a link between the divisors of F 1 F_1 and F 2 F_2 in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack–Kurihara on these elements.

$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic \({\mathbb {Z}}_p\)-extension of K does not admit any proper \(\Lambda \)-submodule of finite index, where \(\Lambda \) is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have \(\Lambda \)-corank one, so they are not \(\Lambda \)-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg–Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley–Lei.

On higher Fitting ideals of certain Iwasawa modules associated with Galois representations and Euler systems

By using “Gauss sum-type” Kolyvagin systems, Kurihara studied the higher Fitting ideals of Iwasawa modules arising from Greenberg Selmer groups of p-adic Galois representations, and proved a refinement of the Iwasawa main conjecture. In this article, we study the higher Fitting ideals of Iwasawa modules arising from the dual fine Selmer groups of general Galois representations which have rank one Euler systems of “Rubin-type” circular units or Beilinson–Kato elements. By using Kolyvagin derivatives, we construct an ascending filtration {Ci(c)}i≥0 of the Iwasawa algebra, and we show that the filtration {Ci(c)}i≥0 gives good approximation of the higher Fitting ideals of the Iwasawa module under the assumption analogous to the Iwasawa main conjecture.