Selmer Groups Of Elliptic Curves Research Articles

Statistics for iwasawa invariants of elliptic curves, II

Statistics for iwasawa invariants of elliptic curves, II

The Selmer groups of elliptic Curves $E_{n}: y^2=x^3+nx$

The Selmer groups of elliptic Curves $E_{n}: y^2=x^3+nx$

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.

The geometric distribution of Selmer groups of elliptic curves over function fields

Fix a positive integer n and a finite field {mathbb {F}}_q. We study the joint distribution of the rank {{,mathrm{rk},}}(E), the n-Selmer group text {Sel}_n(E), and the n-torsion in the Tate–Shafarevich group as E varies over elliptic curves of fixed height d ge 2 over {mathbb {F}}_q(t). We compute this joint distribution in the large q limit. We also show that the “large q, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

Fitting Ideals in Two-variable Equivariant Iwasawa Theory and an Application to CM Elliptic Curves

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.

Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.

Large families of elliptic curves ordered by conductor

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient$\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

Statistics for Iwasawa invariants of elliptic curves

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.

The geometric average size of Selmer groups over function fields

We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over $\mathbb F_q(t)$ is the sum of the divisors of $n$. As a corollary, again in the large $q$ limit, we deduce that $100\%$ of elliptic curves of bounded height over $\mathbb F_q(t)$ have rank $0$ or $1$.

$$\Lambda $$-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.

An analogue of Kida's formula for fine Selmer groups of elliptic curves

In this paper, we prove an analogue of Kida's formula for the fine Selmer groups of elliptic curves. We study the growth behaviour of the fine Selmer groups in p-power degree extensions of the cyclotomic Zp-extension of a totally real number field and obtain a formula for the Zp co-rank of the fine Selmer group of an elliptic curve.

On the Algebraic Functional Equation of the Eigenspaces of Mixed Signed Selmer Groups of Elliptic Curves with Good Reduction at Primes above p

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$. Under suitable assumptions, we prove that the $\eta$-eigenspace and the $\bar{\eta}$-eigenspace of mixed signed Selmer group of the elliptic curve are pseudo-isomorphic. As a corollary, we show that the $\eta$-eigenspace is trivial if and only if the $\bar{\eta}$-eigenspace is trivial. Our results can be thought as a reflection principle which relate an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.

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.

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.

Selmer groups in Iwasawa theory and congruences.

This article outlines the behaviour of Iwasawa μ-invariants for Selmer groups of elliptic curves when the residual representations are equivalent. This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.

On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

AbstractIn this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the “weak main conjecture” à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the “strong main conjecture” suggested by the second named author under the validity of the $\pm $-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among “finite layer objects”, “$\pm $-objects”, and “fine objects” in Iwasawa theory. The case of good ordinary reduction is also treated.

Bounding cubic-triple product Selmer groups of elliptic curves

Let E be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over Q constructed from E through multiplicative induction; it is of rank 8. We show that, under certain assumptions on E, the non-vanishing of the central critical value of the L-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0.

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.

The spin of prime ideals

Fixing a nontrivial automorphism of a number field K, we associate to ideals in K an invariant (with values in {0,±1}) which we call the spin and for which the associated L-function does not possess Euler products. We are nevertheless able, using the techniques of bilinear forms, to handle spin value distribution over primes, obtaining stronger results than the analogous ones which follow from the technology of L-functions in its current state. The initial application of our theorem is to the arithmetic statistics of Selmer groups of elliptic curves.

Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a $D_{2p^{n}}$-extension

We show a p-parity result in a D 2p n -extension of number fields L/K (p≥5) for the twist 1⊕η⊕τ: W(E/K,1⊕η⊕τ)=(-1) 1⊕η⊕τ,X p (E/L) , where E is an elliptic curve over K, η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal (L/K)=D 2p n , and X p (E/L) is the p-Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p-parity conjecture (using the machinery of the Dokchitser brothers).