Selmer Groups Research Articles

Large Selmer groups over number fields

AbstractLet p be a prime number and M a quadratic number field, M ≠ ℚ() if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ℚ with Galois group D2p and an elliptic curve E/ℚ such that F contains M and the p-Selmer group of E/F has size at least pd.

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.

Iwasawa invariants for the False–Tate extension and congruences between modular forms

For an ordinary prime p ⩾ 3 , we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q ( μ p ) whose Galois group is G ≅ Z p ⋊ Z p . For Selmer groups defined over the cyclotomic Z p -extension of Q ( μ p ) , we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of $\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})$ . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

On the Eisenstein ideal for imaginary quadratic fields

AbstractFor certain algebraic Hecke charactersχof an imaginary quadratic fieldFwe define an Eisenstein ideal in ap-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the specialL-valueL(0,χ). We further prove that its index is bounded from above by thep-valuation of the order of the Selmer group of thep-adic Galois character associated toχ−1. This uses the work of R. Tayloret al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms ofL(0,χ), coinciding with the value given by the Bloch–Kato conjecture.

Root numbers, Selmer groups, and non-commutative Iwasawa theory

Let E E be an elliptic curve over a number field F F , and let F ∞ F_\infty be a Galois extension of F F whose Galois group G G is a p p -adic Lie group. The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of E E over F ∞ F_\infty , and the global root numbers attached to the twists of the complex L L -function of E E by Artin representations of G G .

Selmer groups and Tate–Shafarevich groups for the congruent number problem

We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve E_n: y^2 = x^3 − n^2x . We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate–Shafarevich groups and three large Tate–Shafarevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate–Shafarevich groups for square-free positive odd integers n ≤ X is \frac 1 2 \log \log X + O(1) , as X → ∞ .

2-Selmer groups and the Birch–Swinnerton-Dyer Conjecture for the congruent number curves

We take an approach toward counting the number of integers n for which the curve E n : y 2 = x 3 − n 2 x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses on restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the “independence” of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.

Self-duality of Selmer groups

AbstractThe first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the $\Q_p$G-representation naturally associated to the p∞-Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.

INDIVISIBILITY OF ORDERS OF SELMER GROUPS FOR MODULAR FORMS

In this paper, we consider indivisibility of orders of Selmer groups for modular forms under quadratic twists. Then, we will give a generalization of a theorem of James–Ono and Kohnen–Ono.

CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with [Formula: see text] and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups SelE(L)l (l any prime) as L varies through the subextensions of 𝔽 via appropriate versions of Mazur's Control Theorem. In the case l = p, we let 𝔽 = ∪ 𝔽d where 𝔽d/F is a [Formula: see text]-extension. We prove that Sel E(𝔽d)p is a cofinitely generated ℤp[[ Gal (ℤd/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤp[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

Selmer groups for elliptic curves in \mathbb{Z}_l^d-extensions of function fields of characteristic p

Let F be a function field of characteristic p>0, ℱ/F a ℤ l d -extension (for some prime l≠p) and E/F a non-isotrivial elliptic curve. We study the behaviour of the r-parts of the Selmer groups (r any prime) in the subextensions of ℱ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of ℱ/F.

Erratum for: “On the parity of ranks of Selmer groups. III” [cf. Documenta Math. 12 (2007), 243–274

In the manuscript “On the Parity of Ranks of Selmer Groups III” Documenta Math. 12 (2007), 243–274, [1], Remark 4. 1.2(4) and the treatment of archimedean \varepsilon -factors in 4.1.3 are incorrect. Contrary to what is stated in 0.3, the individual archimedean \varepsilon -factors \varepsilon_u(M) (u\mid\infty) cannot be expressed, in general, in terms of M_{\mathfrak{p}} , but their product can.

Generalised Euler characteristics of Selmer groups

Let E be an elliptic curve defined over a number field F, and let p ⩾ 5 be a prime. In this paper, we study the structure of the Selmer group of E, over p-adic Lie extensions F∞/F. In particular, under certain global and local conditions on F∞ we relate the generalised Gal(F∞/F)-Euler characteristic of Sel(E/F∞) to the generalised Euler characteristic of the Selmer group over the cyclotomic ℤp-extension of F. This invariant generalises the classical Euler characteristic to the case when rankℤE(F) > 0. Moreover, we show that the global and local conditions on F∞ are satisfied for a large class of p-adic Lie extensions of F.

The symmetric structure of the plus/minus Selmer groups of elliptic curves over totally real fields and the parity conjecture

By improving the techniques of [B.D. Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007) 47–72] we prove some symmetric structure of the minus Selmer groups of elliptic curves for supersingular primes. This structure was already known for the Selmer groups for ordinary primes [J. Nekovar, On the parity of ranks of Selmer groups II, C. R. Math. Acad. Sci. Paris Ser. I 332 (2) (2001) 99–104; J. Nekovar, Selmer complexes, Astérisque 310 (2006)]. One consequence is the parity conjecture over a totally real field under some conditions.

The effect of twisting on the 2-Selmer group

AbstractLet Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

Nonsmooth classical points on eigenvarieties

We construct nonsmooth points on unitary eigenvarieties. More precisely, we construct points so that the local ring of the irreducible components of the eigenvariety through this point is nonsmooth and not even a unique factorization domain (UFD). We use those points to construct geometrically several independent extensions in the relevant Selmer group

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.

Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich–Tate group

Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over Q and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly approximate Heegner points over ring class fields and use these points to give evidence for the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two then the Z p -corank of the corresponding Selmer group is at least two as well. We also use explicitly computed Heegner points to produce non-trivial classes in the Shafarevich–Tate group.

Distribution of Selmer groups of quadratic twists of a family of elliptic curves

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q . We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate–Shafarevich group and a large Tate–Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1–2) (2005) 139–154 (2006)], this shows that the mean value of the 2-rank of the large Tate–Shafarevich group for square-free positive integers n less than X is 1 2 log log X + O ( 1 ) , as X → ∞ .