Dual Selmer Group Research Articles

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 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.

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.

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.

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 SELMER GROUP OF A CERTAIN -ADIC LIE EXTENSION

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

Stark systems over Gorenstein local rings

In this paper, we define a module of Stark systems over a complete Gorenstein local ring with a finite residue field of odd characteristic. Under some mild assumptions, we show that it is free of rank one and that these systems control all the Fitting ideals of the Pontryagin dual of the dual Selmer group. These results generalize the work of B. Mazur and K. Rubin on Stark systems over a principal ideal ring.

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.

A remark on the 𝔐H(G)-conjecture and Akashi series

In this paper, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above p to satisfy the 𝔐H(G)-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the 𝔐H(G)-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series. We will apply this criterion to prove some results on the characteristic elements of the dual Selmer groups. Our methods in this paper are inspired by the work of Coates–Schneider–Sujatha and can be extended to the Greenberg Selmer groups attached to other ordinary representations, for instance, those coming from a p-ordinary modular form.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

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 bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate–Shafarevich group of an elliptic curve in the Z p 2 -extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

Pairings and functional equations over the GL2 -extension†

We construct a pairing on the dual Selmer group over the GL 2-extension ℚ(E[p∞]) of an elliptic curve without complex multiplication and with good ordinary reduction at a prime p ⩾ 5 whenever it satisfies certain, conjectured, torsion properties. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we reduce the parity conjecture for the p-Selmer rank and the analytic root number for the twists of elliptic curves with self-dual Artin representation to the case when the Artin representation factors through the quotient of Gal (ℚ(E[p∞])/ℚ) by its maximal pro-p normal subgroup. This gives a new proof of the parity conjecture whenever the elliptic curve E has a p-isogeny over the rationals.

ELLIPTIC CURVES RELATED TO CYCLIC CUBIC EXTENSIONS

The aim of this paper is to study certain family of elliptic curves [Formula: see text] defined over a number field F arising from hyperplane sections of some cubic surface [Formula: see text] associated to a cyclic cubic extension K/F. We show that each [Formula: see text] admits a 3-isogeny ϕ over F and the dual Selmer group [Formula: see text] is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve [Formula: see text] with nice property. We also prove that the Shafarevich–Tate group [Formula: see text] coincides with a class group of K as a special case.

Lifting n-Dimensional Galois Representations

AbstractWe investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over ℚ of certain infinite subgroups of p-adic general linear groups.

Characteristic elements, pairings and functional equations over the false Tate curve extension

AbstractWe construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a primep≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of thep-adicL-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

Analogue of Kida's formula for certain strongly admissible extensions

We study the Iwasawa theory of elliptic curves over certain infinite (non-commutative) p-adic Galois–Lie extensions. In particular, we consider the analogue of the classical Iwasawa λ-invariant and Kida's formula for the dual Selmer group.

SELMER GROUPS AND CHOW GROUPS OF SELF-PRODUCTS OF ALGEBRAIC VARIETIES

Let X be a proper flat scheme over the ring of integers of a global field. We show that the Tate conjecture and the finiteness of the Chow group of vertical cycles on self-products of X implies the vanishing of the dual Selmer group of certain twists of tensor powers of representations occurring in the étale cohomology of X.