A classification of Q-curves with complex multiplication

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

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

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factoriza- tion of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K ' , when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in Bp,∞, the definite quaternion algebra over Q ramified only at p and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo p|p between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields, and to give a bound on those supersingular primes which can appear. In the special case of Jacobians of genus 2 curves, this provides information about the factorizations of numerators of Igusa invariants, and so is also relevant to the problem of constructing genus 2 curves for use in cryptography.

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Rational points on elliptic curves with complex multiplication by the ring of integers in [formula omitted](√−7)

Given an elliptic curve E/k and a Galois extension k ' /k, we construct an exact functor from torsion-free modules over the endomorphism ring EndE k ' with a semilinear Gal(k ' /k) action to abelian varieties over k that are k ' -isogenous to a power of E. As an application, we give a simple proof that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.

Let $E$ be an elliptic curve defined over $\Q$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We apply an elliptic unit version of Ichimura–Sumida criterion for the $\Z_p$-extension $F_\infty/F$ associated to $E$ and try to determine the characteristic polynomial of the maximal unramified abelian $p$-extension of $F_\infty$.

The j-invariants of the quadratic Q-curves without complex multiplication are studied. Some properties of the norms of these invariants are shown and a relationship between the field Q(j) and the degree of an isogeny of the Q-curve to its Galois conjugate is found. In the case when the degree of the isogeny is a prime p, some properties of the primes of potentially multiplicative reduction for the Q-curve and of the reduction of j modulo a prime P in Q(j) over p when the Q-curve has potentially good reduction at P are found.

The goal of this article is to give an explicit classification of the\npossible $p$-adic Galois representations that are attached to elliptic curves\n$E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an\nimaginary quadratic field, and let $\\mathcal{O}_{K,f}$ be an order in $K$ of\nconductor $f\\geq 1$. Let $E$ be an elliptic curve with CM by\n$\\mathcal{O}_{K,f}$, such that $E$ is defined by a model over\n$\\mathbb{Q}(j(E))$. Let $p\\geq 2$ be a prime, let $G_{\\mathbb{Q}(j(E))}$ be the\nabsolute Galois group of $\\mathbb{Q}(j(E))$, and let $\\rho_{E,p^\\infty}\\colon\nG_{\\mathbb{Q}(j(E))}\\to \\operatorname{GL}(2,\\mathbb{Z}_p)$ be the Galois\nrepresentation associated to the Galois action on the Tate module $T_p(E)$. The\ngoal is then to describe, explicitly, the groups of\n$\\operatorname{GL}(2,\\mathbb{Z}_p)$ that can occur as images of\n$\\rho_{E,p^\\infty}$, up to conjugation, for an arbitrary order\n$\\mathcal{O}_{K,f}$.\n

The growth of the Mordell-Weil rank of an elliptic curve in a tower of number fields can be discussed at many levels.On the one hand, the issues raised can be embedded in a broader Iwasawa theory of elliptic curves (Mazur [10]); on the other hand, they can be crystallized in a single easily stated question, namely whether the rank of the elliptic curve over subextensions of finite degree is bounded in the tower.But whatever one's point of view, the case of abelian towers has seen important advances in recent years, including notably the results of Kato on cyclotomic towers, of Cornut [2] and Vatsal [18] on anticyclotomic towers, and of Skinner and Urban on Mazur's conjecture.In the case of nonabelian towers, by contrast, virtually nothing is known, and even a conjectural framework has begun to emerge only recently [1].However Greenberg [7] observed more than twenty years ago that some insight could already be gained from the classical conjectures about L-functions: They imply that for certain pairs (E 1 , E 2 ) of elliptic curves of relatively prime conductor there exist primes p such that the rank of E 1 is unbounded in the p-division tower of E 2 (which can be chosen not to have complex multiplication, so that the division tower is nonabelian).Greenberg's idea was elaborated further by L. Howe [8], who gave a conjectural lower bound for the growth of the rank of E 1 in the division tower of E 2 .The aim of the present note is to show that the standard conjectures also have some bearing on the growth of the rank of an elliptic curve in its own division tower.For example, we shall see that if E is a semistable elliptic curve over Q and p is a sufficiently large prime congruent to 3 mod 4 then the rank of E should be unbounded in its p-division tower.As in the papers of Greenberg and Howe, the mechanism for arriving at such conclusions is a root number calculation.Fix a number field F , an elliptic curve E over F , and a prime p, and put, where E[p ∞ ] denotes the group of points on E of p-power order.The root numbers at issue here are associated to the L-functions L(s, E, τ ), where τ runs over irreducible self-dual representations of Gal(F ∞ /F ).For a precise definition of L(s, E, τ ) we refer the reader to [12], pp.151 and 156, or to [1], §5, but we should at least spell out our group-theoretic conventions: First of all, a representation of a topological group is understood to be continuous, finite-dimensional, and defined over the complex numbers.In particular, in the case of a profinite group like Gal(F ∞ /F ) a representation is trivial on an open subgroup and hence factors through a finite quotient.It also follows that when the term character is used in the sense of "one-dimensional representation" we are

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication byi. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2= − 2.

Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form ∑ k = 0 ∞ ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.