Lie Extensions Research Articles

On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields

We establish the Iwasawa main conjecture for semistable abelian varieties over a function field of characteristic p under certain restrictive assumptions. Namely we consider p -torsion free p -adic Lie extensions of the base field which contain the constant \mathbb{Z}_p -extension and are everywhere unramified. Under the usual \mu=0 hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of p -adic cohomology [F. Trihan, D. Vauclair, A comparison theorem for semi abelian schemes over a smooth curve, preprint arXiv:1505.02942, 2015] together with the trace formulas of J.-Y. Etesse and B. Le Stum [Math. Ann. 296, No. 3, 557–576 (1993; Zbl 0789.14015)].

On the Iwasawa asymptotic class number formula for $\mathbb {Z}_{p}^r\rtimes \mathbb {Z}_{p}$-extensions

Let $p$ be an odd prime and $F_{\infty,\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group isomorphic to $\mathbb{Z}_p^r\rtimes\mathbb{Z}_p$, $r\geq 1$. Under certain assumptions, we prove an asymptotic formula for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. This generalizes a previous result of Lei, where he establishes such a formula in the case $r=1$. An important and new ingredient towards extending Lei's result rests on an asymptotic formula for a finitely generated (not necessarily torsion) $\mathbb{Z}_p[[\mathbb{Z}_p^r]]$-module which we will also establish in this paper. We then continue studying the growth of $p$-exponents of the class groups under more restrictive assumptions and show that there is an asymptotic formula in our noncommutative $p$-adic Lie extension analogous to a refined formula of Monsky (which is for the commutative extension) in a special case.

Euler characteristic and Akashi series for Selmer groups over global function fields

Let A be an abelian variety defined over a global function field F of positive characteristic p and let K/F be a p-adic Lie extension with Galois group G. We provide a formula for the (truncated) Euler characteristic χ(G,SelA(K)p) of the p-part of the Selmer group of A over K. In the special case G≃Zpd and A a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of SelA(K)p is related to special values of a p-adic L-function.

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Let $p$ be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian $p$-adic Lie extensions. Crucially, this result does not depend on the vanishing of the relevant Iwasawa $\mu$-invariant.

Fine Selmer groups of congruent Galois representations

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible p-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the π-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-p extension of an admissible p-adic Lie extension and the structure of the dual fine Selmer group over the said admissible p-adic Lie extension.

Wach Modules, Regulator Maps, and $\varepsilon$-Isomorphisms in Families

Abstract We prove the “local $\varepsilon$-isomorphism” conjecture of Fukaya and Kato [13] for certain crystalline families of $G_{{\mathbf{Q}_p}}$-representations. This conjecture can be regarded as a local analog of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-$1$ modules (cf. [33]), of Benois and Berger for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to the cyclotomic extension (cf. [1]), as well as of Loeffler et al. (cf. [21]) for crystalline $G_{{\mathbf{Q}_p}}$-representations with respect to abelian $p$-adic Lie extensions of ${\mathbf{Q}_p}$. Nakamura [24, 25] has also formulated a version of Kato’s $\varepsilon$-conjecture for affinoid families of $(\varphi,\Gamma)$-modules over the Robba ring, and proved his conjecture in the rank-$1$ case. He used this case to construct an $\varepsilon$-isomorphism for families of trianguline $(\varphi,\Gamma)$-modules, depending on a fixed triangulation. Our results imply that this $\varepsilon$-isomorphism is independent of the chosen triangulation for certain crystalline families. The main ingredient of our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger [6] and following Kisin’s approach to the construction of potentially semi-stable deformation rings [18].

Comparing the $\pi$-primary submodules of the dual Selmer groups

In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $\pi$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are Tate dual to each other are pseudo-isomorphic (up to a twist of the Iwasawa algebra). In the second situation, our result will be applied to study the variation of the $\pi$-primary submodules of the dual Selmer groups of certain specialization of a big Galois representation. One of the important ingredient in our proofs is an asymptotic formula for $\pi$-primary modules over a noncommutative Iwasawa algebra which can be viewed as a generalization of a weak analog of the classical Iwasawa asymptotic formula.

Estimating class numbers over metabelian extensions

Let $p$ be an odd prime and $L/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to $L$, we study the asymptotic behaviours of the size of the $p$-primary part of the ideal class groups over certain finite subextensions inside $L/K$. This generalizes the classical result of Iwasawa and Cuoco-Monsky in the abelian case and gives a more precise formula than a recent result of Perbet in the non-commutative case when $d=2$.

Multivariable (φ,Γ)-modules and locally analytic vectors

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic means that $\Gamma={\rm Gal}(K_\infty/K)$ where $K_\infty$ is the cyclotomic extension of $K$). One particularly convenient feature of the cyclotomic theory is the fact that any $(\varphi,\Gamma)$-module is overconvergent. Questions pertaining to the $p$-adic local Langlands correspondence lead us to ask for a generalization of the theory of $(\varphi,\Gamma)$-modules, with the cyclotomic extension replaced by an infinitely ramified $p$-adic Lie extension $K_\infty / K$. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely the case of a Lubin-Tate extension, most $(\varphi,\Gamma)$-modules fail to be overconvergent. In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of $\Gamma$ inside some big modules defined using Fontaine's rings of periods. We show that, in the cyclotomic case, we recover the ususal overconvergent $(\varphi,\Gamma)$-modules. In the Lubin-Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that $(\varphi,\Gamma)$-modules attached to $F$-analytic representations are overconvergent.

Ramification filtrations of certain abelian Lie extensions of local fields

Let G⊂xFq〚x〛 (q is a power of the prime p) be a subset of formal power series over a finite field such that it forms a compact abelian p-adic Lie group of dimension d≥1. We establish a necessary and sufficient condition for the APF extension of local field corresponding to (Fq⸨x⸩,G) under the field of norms functor to be an extension of p-adic fields. We then apply this result to study invertible power series over a ring of p-adic integers which commute with a fixed noninvertible power series under composition.

Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Let $$E_{/\mathbb {Q}}\!$$ be an elliptic curve, $$p>3$$ a good ordinary prime for E, and $$K_\infty $$ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions $$K_n/k$$ inside $$K_\infty $$ ; we obtain lower bounds on the former, and upper bounds on the latter’s size.

Akashi series, characteristic elements and congruence of Galois representations

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible [Formula: see text]-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain a similar result for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.

Control theorems for $\ell$-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0, K/F an l-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety. We study SelA(K) ∨ (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Zl((Gal(K/F)))-module via generalizations of Mazur's Control Theorem. If Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H ≃ Zl, we are able to give sufficient conditions forSelA(K) ∨ to be finitely generated as Zl((H))-module and, consequently, a torsion Zl((Gal(K/F)))-module. We deal with both cases l 6 p and l = p.

On the pseudo-nullity of the dual fine Selmer groups

In this paper, we will study the pseudo-nullity of the dual fine Selmer group and its related question. We investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over an admissible p-adic Lie extension F∞ from the knowledge of the pseudo-nullity of the Galois group of the maximal abelian unramified pro-p extension of F∞ at which every prime of F∞ above p splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside p. We will illustrate our results with many examples.

Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

AbstractLet$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

Growth of in towers for isogenous curves

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.

Algebraic functional equations and completely faithful Selmer groups

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

Local epsilon isomorphisms

In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou regulator map, or its extension to the 2-variable case due to the first and third authors.

On completely faithful Selmer groups of elliptic curves and Hida deformations

In this paper, we study completely faithful torsion Zp〚G〛-modules with applications to the study of Selmer groups. Namely, if G is a nonabelian group belonging to certain classes of polycyclic pro-p group, we establish the abundance of faithful torsion Zp〚G〛-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such Zp〚G〛-modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to have appeared in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we will prove “control theorem” results on the faithfulness of Selmer groups over a general strongly admissible p-adic Lie extension.

On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ℓ-adic Lie extension (ℓ ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ℓ. We shall prove that, under certain conditions, Sel A (K) ℓ ∨ has no nontrivial pseudo-null submodule.