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Non-monogenic Division Fields of Elliptic Curves
Abstract
For various positive integers n, we show the existence of infinite families of elliptic curves over Q with n-division fields that are not monogenic, i.e., such that the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with an algorithm based on ideas of Dedekind. As a counterpoint, we are able to use different aspects of the arithmetic of elliptic curves to exhibit a family of monogenic 2-division fields.
... The n-division fields of elliptic curves Q(E[n]) are a natural class of examples to consider next. This is the subject of [47], where the author shows that there are many division fields of elliptic curves over Q that are not monogenic. ...
... that are not present in dimension one. In spite of this we are able to generalize the main result of [47] to abelian varieties with irreducible Weil polynomials and minimal endomorphism rings (Theorem 5.2). We classify when primes not dividing n are local obstructions to monogenicity. ...
... We begin with a very abbreviated list of some related research for elliptic curves. As mentioned above, [47] is the dimension one predecessor to this work. In [27], González-Jiménez and Lozano-Robledo classify the possible abelian division fields. ...
Let A be an abelian variety over a finite field k with |k| = q = pm. Let π ∈Endk(A) denote the Frobenius and let v = qπ−1 denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When Endk(A) = ℤ[π,v], we construct a matrix which describes the action of π on the prime-to-p-torsion points of A. We employ this matrix in an algorithm that detects when p is an obstruction to the monogenicity of division fields of certain abelian varieties.
... A natural next example to consider is the n-division fields of an elliptic curve. This is the subject of [Smi21], where the author shows that there are many division fields of elliptic curves over Q that are not monogenic. ...
... As is often the case, we encounter phenomena that are not present in dimension one. In spite of this we are able to generalize the main result of [Smi21] to abelian varieties with irreducible Weil polynomials and minimal endomorphism rings (Theorem 5.2). In the process we construct a matrix which yields the action of a lift of the Frobenius at p on the prime-to-p-torsion points of the relevant abelian variety (Theorem 4.1). ...
... actually shows that the ideal class group of Z[π, v] acts freely and transitively on isomorphism classes of abelian varieties over F p with endomorphism ring equal to Z[π, v]. We note that the case where A is a supersingular elliptic curve, k = F p , and π = ± √ p has already been investigated in [Smi21]. ...
Let A be an abelian variety over a finite field k with . Let denote the Frobenius and let denote Verschiebung. Suppose the Weil q-polynomial of A is irreducible. When , we construct a matrix which describes the action of on the prime-to-p-torsion points of A. We employ this matrix in an algorithm that detects when p is an obstruction to the monogeneity of division fields of certain abelian varieties.
... Moreover, for each N , they classified the possible abelian Galois groups that occur for a division field. The classification of abelian division fields of elliptic curves over Q has numerous applications, for example in: the classification of torsion subgroups of elliptic curves [4], [9], [5], [6]; the classification of isogeny-torsion graphs [3], [2]; Brauer groups [21]; non-monogenic number fields [19], [20]; congruences between elliptic curves [7]; or the classification of Galois representations [11], [16], among others. ...
Let K be an imaginary quadratic field, and let be an order in K of conductor . Let E be an elliptic curve with CM by , such that E is defined by a model over , where . In this article, we classify the values of and the elliptic curves E such that (i) the division field is an abelian extension of , and (ii) the N-division field coincides with the N-th cyclotomic extension of the base field.
... To this end, Adelmann's book [1] provides a nice introduction culminating in criteria describing the decomposition of unramified primes in various division fields. An abbreviated list of other relevant work in this area includes [8,9,[18][19][20]26,28,30,41]. ...
Consider an elliptic curve E over a number field K. Suppose that E has supersingular reduction at some prime p of K lying above the rational prime p. We completely classify the valuations of the pn-torsion points of E by the valuation of a coefficient of the pth division polynomial. This classification corrects an error in earlier work of Lozano-Robledo. As an application, we find the minimum necessary ramification at p in order for E to have a point of exact order pn. Using this bound we show that sporadic points on the modular curve X1(pn) cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to X1(N) with N composite.
... In [33], Smith studied the monogenity of radical extensions and gave sufficient conditions for a Kummer extension Q(ξ n , n √ α) to be not monogenic. He also gave in [32] infinite parametric families of non-monogenic division fields of elliptic curves. The question of monogenity in the relative case has been profoundly studied by Győry in [23]. ...
Let K be a nonic number field generated by a complex root of a monic irreducible trinomial , where . Let i(K) be the index of K. A rational prime p dividing i(K) is called a prime common index divisor of K. In this paper, for every rational prime p, we give necessary and sufficient conditions depending only a and b for which p is a common index divisor of K. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given.
... Indeed, Jones [16] proved that 100% of elliptic curves over ℚ are Serre curves in a suitable sense of density. Serre curves have many applications (e.g., [1,2,3,15,17,23,36,41]) largely because the adelic image of such a curve is readily known (it depends only on the discriminant Δ of ). ...
We consider elliptic curves for which the image of the adelic Galois representation is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their -adic images, compute all examples of conductor at most 500,000, precisely describe the image of , and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.
Let be an imaginary quadratic field, and let be the order in of conductor . Let be an elliptic curve with complex multiplication by , such that is defined by a model over , where . In this article, we classify the values of and the elliptic curves such that (i) the division field is an abelian extension of , and (ii) the ‐division field coincides with the th cyclotomic extension of the base field.
We consider elliptic curves E / Q E / \mathbb {Q} for which the image of the adelic Galois representation ρ E \rho _E is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their ℓ \ell -adic images, compute all examples of conductor at most 500,000, precisely describe the image of ρ E \rho _E , and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.
We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial 3-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic fields given by for such that are squarefree, are monogenic.
Let E be an elliptic curve over Q, and let n=>1. The central object of study
of this article is the division field Q(E[n]) that results by adjoining to Q
the coordinates of all n-torsion points on E(Q). In particular, we classify all
curves E/Q such that Q(E[n]) is as small as possible, that is, when
Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5.
More generally, we classify all curves such that Q(E[n]) is contained in a
cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem),
when Q(E[n])/Q is an abelian extension. In particular, we prove that this only
happens for n=2,3,4,5,6, or 8, and we classify the possible Galois groups that
occur for each value of n.
Let E be an elliptic curve over a finite field k, and ℓ a prime number different from the characteristic of k. In this paper, we consider the problem of finding the structure of the Tate module Tℓ(E) as an integral Galois representations of k. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial fE(x) and the j-invariant jE of E. Hilbert Class Polynomials of imaginary quadratic orders play an important role here. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.
In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of such a division field. Such fields include a large class of non-solvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the elliptic curve reduced modulo p.
The aim of this book is to present an exposition of the theory of alge braic numbers, excluding class-field theory and its consequences. There are many ways to develop this subject; the latest trend is to neglect the classical Dedekind theory of ideals in favour of local methods. However, for numeri cal computations, necessary for applications of algebraic numbers to other areas of number theory, the old approach seems more suitable, although its exposition is obviously longer. On the other hand the local approach is more powerful for analytical purposes, as demonstrated in Tate's thesis. Thus the author has tried to reconcile the two approaches, presenting a self-contained exposition of the classical standpoint in the first four chapters, and then turning to local methods. In the first chapter we present the necessary tools from the theory of Dedekind domains and valuation theory, including the structure of finitely generated modules over Dedekind domains. In Chapters 2, 3 and 4 the clas sical theory of algebraic numbers is developed. Chapter 5 contains the fun damental notions of the theory of p-adic fields, and Chapter 6 brings their applications to the study of algebraic number fields. We include here Shafare vich's proof of the Kronecker-Weber theorem, and also the main properties of adeles and ideles.
Given an elliptic curve , the torsion points of E give rise to a natural Galois representation associated to E. In 1972, Serre showed that for all non-CM elliptic curves. The main goal of this paper is to exhibit an elliptic surface such that the Galois representations associated to almost all of the rational specializations have maximal image. Further, we find an explicit set , such that if , then the Galois representation associated to the specialization at t has maximal image, with a bounded number of exceptions.
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points.
For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.
The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.
Let E be an elliptic curve over ℚ and let ℚ(E[n]) be its nth division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of ℚ(E[n])/ℚ is as large as possible, that is, GL2(ℤ/nℤ), for all integers n coprime to a constant integer m(E, ℚ) depending (at most) on E/ℚ. Serre also showed that the best one can hope for is to have |GL2(ℤ/nℤ) : Gal(ℚ(E[n])/ℚ)| ≤ 2 for all positive integers n. We study the frequency of this optimal situation in a one-parameter family of elliptic curves over ℚ, and show that in essence,
for almost all one-parameter families, almost all elliptic curves have this optimal behavior.
Let m be any positive integer, R be a ring with identity, M m (R) be the matrix ring of all m by m matrices over R and G m (R) be the multiplicative group of all m by m nonsingular matrices in M m (R). In this paper, the following are investigated: (1) For any pairwise coprime ideals {I 1 ,I 2 ,⋯,I n } in a ring R, M m (R/(I 1 ∩I 2 ∩⋯∩I n )) is isomorphic to M m (R/I 1 )×M m (R/I 2 )×⋯×M m (R/I n ), and so G m (R/(I 1 ∩I 2 ∩⋯∩I n )) is isomorphic to G m (R/I 1 )×G m (R/I 2 )×⋯×G m (R/I n ); (2) In particular, if R is a finite ring with identity, then the order of G m (R) can be computed.
Let K be an imaginary quadratic field other than
and . We construct relative power integral bases between
certain abelian extensions of K in terms of Weierstrass units.
2nd Ed., Substantially Rev. and Extended
Using a multidimensional large sieve inequality, we obtain a bound for the mean square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove a theorem to the effect that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.
Unités modulaires et monogénéité d'anneaux d'entiers
- Cassou-Noguès
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- V Radhakrishnan
Volume I: Contributions to Number Theory
- Kummer