Selmer Groups Of Elliptic Curves Research Articles

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).

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.

ON THE 2-ADIC IWASAWA INVARIANTS OF ORDINARY ELLIPTIC CURVES

In this paper, we give an explicit formula describing the variation of the 2-adic Iwasawa λ-invariants attached to the Selmer groups of elliptic curves under quadratic twists. To prove this formula, we extend some results known for odd primes p, an analogue of Kida's formula proved by Hachimori and the author and a formula given by Greenberg and Vatsal, to the case where p = 2.

CENTRES OF SKEWFIELDS AND COMPLETELY FAITHFUL IWASAWA MODULES

Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebraH of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebraG of G := H × Zp. We also show that a finitely generated torsionG-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

Average size of $2$-Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over Q with Q-torsion group Z 2 × Z 2 , and prove that the average order of the 2-Selmer groups is bounded.

Fine Selmer groups of elliptic curves over p-adic Lie extensions

The aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic Iwasawa theory. Let F be a finite extension of Q, p an odd prime number, and F cyc the cyclotomic Zp-extension of Q. Let K(F cyc) denote the maximal unramified abelian p-extension of F cyc, in which every prime of F cyc above p splits completely, and define Y (F cyc) to be the Galois group of K(F cyc) over F cyc. Put = G(F cyc/F ), and write ( ) for the Iwasawa algebra of . Iwasawa proved that Y (F cyc) is always a finitely generated torsion ( )-module, and he conjectured [12] that in fact Y (F cyc) is always a finitely generated Zp-module. At present, this conjecture has only been proven when F is abelian over Q (see [5],[26]). Perhaps surprisingly, it does not seem to have been pointed out in the literature that there is a precise analogue of Iwasawa’s conjecture for elliptic curves over F cyc. Let E be an elliptic curve over F , and let

Selmer Groups of Elliptic Curves with Complex Multiplication

AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

The Selmer Groups of Elliptic Curves

Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map λ K and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X 0( p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p⩽7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.

On the parity of ranks of Selmer groups II

We prove the parity conjecture for Selmer groups of elliptic curves over Q in the case of good ordinary reduction.

On finite $\Lambda $-submodules of Selmer groups of elliptic curves

In this note, we give another proof of a result of R. Greenberg on the non-existence of non-trivial finite $\Lambda$-submodules of Selmer groups.

Abelian varieties and the class invariant homomorphism

We discuss the problem of a generalization to higher dimensions of an earlier result of ours on a geometrical description of Selmer groups of elliptic curves over global fields of positive characteristic.

Estimating the 2-rank of Cubic Fields by Selmer Groups of Elliptic Curves

Frey and his coauthors have established a relationship between the 2-torsion of the Selmer group of an elliptic curve of the special formE:y2=x3±k2and the 2-class number of pure cubic fieldK=Q((∓k2)1/3)=Q((∓k)1/3).In the present paper we prove a far-reaching generalization of an analogous relationship between the 2-rank of any non-Galois cubic number fieldKand the 2-torsion of the Selmer group of a corresponding elliptic curve. We implemented the resulting algorithm and used it, e.g., to produce four cubic number fields of exact 2-rank 7. The 2-rank of number fields is of special interest because if it is sufficiently large the number field has an infinite class field tower. In particular, the four fields of 2-rank 7 turn out to have infinite class field towers.

The selmer groups of elliptic curves and the ideal class groups of quadratic fields

Let D be an integer. Consider the elliptic curve E/Q :y2 = x3 + D, which has j-invariant 0. We can show that for this elliptic curve the rank of its 3-Selmer group is closely related to the 3-rank of the ideal class groups of the quadratic fields and . Fol the same family of curves Frey showed that, if D is a cube, the rank of the Selrner group of a 3-isogeny is related to the class number of the quadratic field [3]. Also Jan Nekeváȓ proved some analogous result for elliptic curve given by Dy2 = 4x3 − 27 which is isomorphic to the curve given by y2 = x3 − 432D3 [4]. Our method is different from theirs and it can give a far more complete result for general D.