Lie Extensions Research Articles

Structure of fine Selmer groups over -extensions

AbstractThis 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.

Control Theorems for Fine Selmer Groups

We study the growth of the p-primary fine Selmer group, R(E/F ′ ), of an elliptic curve over an intermediate sub-extension F ′ of a p-adic Lie extension, ℒ/F. We estimate the ℤ p -corank of the kernel and cokernel of the restriction map r ℒ/F ′ :R(E/F ′ )→R(E/ℒ) Gal(ℒ/F ′ ) with F ′ a finite extension of F contained in ℒ. We show that the growth of the fine Selmer groups in these intermediate sub-extension is related to the structure of the fine Selmer group over the infinite level. On specializing to certain (possibly non-commutative) p-adic Lie extensions, we prove finiteness of the kernel and cokernel and provide growth estimates on their orders.

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 the growth of even K-groups of rings of integers in p-adic Lie extensions

Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficients in $\mathbb{Z}_p(i)$ for $i\geq 2$. We should mention that this latter approach is possible thanks to the Quillen-Lichtenbaum Conjecture which is now known to be valid by the works of Rost-Voevodsky. We also note that the approach allows us to work with more general $p$-adic Lie extensions that do not necessarily contain the cyclotomic $\mathbb{Z}_p$-extension, where the Kummer theoretical approach does not apply. Along the way, we study the torsionness of the second Iwasawa cohomology groups with coefficients in $\mathbb{Z}_p(i)$ for $i\geq 2$. Finally, we give examples of $p$-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial $\mu$-invariants.

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.

On order of vanishing of characteristic elements

AbstractLetpbe a fixed odd prime. LetEbe an elliptic curve defined over a number fieldFwith either good ordinary reduction or multiplicative reduction at each prime ofFabovep. We shall study the characteristic element of the Selmer group ofEover ap-adic Lie extension. In particular, we relate the order of vanishing of these characteristic elements evaluated at Artin representations to the Selmer coranks and their twists in the intermediate subextensions of thep-adic Lie extension.

A Dynamical Analogue of Sen’s Theorem

Abstract We study the higher ramification structure of dynamical branch extensions and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen’s theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse–Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger [8] and a partial answer to a question about wild ramification in arboreal extensions of number fields [1, 9].

Arithmetic statistics and noncommutative Iwasawa theory

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 \mathrm{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.

The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture

Let K denote a number field containing a primitive p -th root of unity; if p=2 , then we assume K to be totally imaginary. If K_\infty/K is a \mathbb{Z}_p -extension such that no prime above p splits completely in K_\infty/K , then the vanishing of Iwasawa's invariant \mu(K_\infty/K) implies that the weak Leopoldt Conjecture holds for K_\infty/K . This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible p -adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible p -adic Lie extensions.

On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.

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.

On p-class groups of relative cyclic p-extensions

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

Twisting lemma for $\lambda$-adic modules

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[\Gamma ]]$ with $\Gamma \cong \mathbb{Z}_p, \exists$ a continuous character $\theta: \Gamma \rightarrow \mathbb{Z}_p^\times$ such that, the $ \Gamma^{n}$-Euler characteristic of the twist $M(\theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $\mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $\mathcal{T}$ is a finite extension of $\mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $\Lambda$-adic form over a $p$-adic Lie extension.

Control theorem and functional equation of Selmer groups over p-adic Lie extensions

Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even in the case where the Selmer group is associated to an elliptic curve. We also discuss the functional equation for the analytic $p$-adic $L$-functions and check the compatibility with the functional equation of the dual Selmer groups.

Growth of Selmer groups and fine Selmer groups in uniform pro-p extensions

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

On 2-adic Lie Iterated Extensions of Number Fields Arising from a Joukowski Map

We construct a special $2$-adic Lie extension of a number field as an iterated tower by a conjugate of Joukowski map. If the number field is totally real, the unramified Iwasawa module over such a $2$-adic Lie iterated extension is conjecturally pseudo-null under Greenberg's conjecture for all intermediate cyclotomic $\mathbb{Z}_2$-extensions. We give some examples of such $2$-adic Lie iterated extensions with pseudo-null Iwasawa modules.

Euler characteristics and their congruences in the positive rank setting

AbstractThe notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

On the growth of Mordell–Weil ranks in $p$-adic Lie extensions

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of the abelian variety of $A$ in the said $p$-adic Lie extension. Our upper bound can be expressed in terms of invariants coming from the cyclotomic level. Motivated by this formula, we make a conjecture on an asymptotic upper bound of the growth of Mordell-Weil ranks over a $p$-adic Lie extension which is in terms of the Mordell-Weil rank of the abelian variety over the cyclotomic $\mathbb{Z}_p$-extension. Finally, it is then natural to ask whether there is such a conjectural upper bound when the abelian variety has non-ordinary reduction. For this, we can at least modestly formulate an analogue conjectural upper bound for the growth of Mordell-Weil ranks of an elliptic curve with good supersingular reduction at the prime $p$ over a $\mathbb{Z}_p^2$-extension of an imaginary quadratic field.

A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish asymptotic upper bounds for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that $H\cong \mathbb{Z}_p$.