CM Elliptic Curves Research Articles

Orders of CM elliptic curves modulo p with at most two primes

Nowadays the generation of cryptosystems requires two main aspects. First the security, and then the size of the keys involved in the construction and comunication process. About the former one needs a di±cult mathematical assumption which ensures your system will not be broken unless a well known di±cult problem is solved. In this context one of the most famous assumption underlying a wide variety of cryptosystems is the computation of logarithms in ¯nite ¯elds and the Di±e Hellman assumption. However it is also well known that elliptic curves provide good examples of representation of abelian groups reducing the size of keys needed to guarantee the same level of security as in the ¯nite ¯eld case. The ¯rst thing one needs to perform elliptic logarithms which are computationaly secure is to ¯x a ¯nite ¯eld, Fp, and one curve, E=Fp de¯ned over the ¯eld, such that jE(Fp)j has a prime factor as large as possible. In practice the problem of ¯nding such a pair, of curve and ¯eld, seems simple, just take a curve with integer coe±cients and a prime p of good reduction at random and see if jE(Fp)j has a big prime factor. However the theory that makes the previous algorithm useful is by no means obvious, neither clear or complete. For example it is well known that supersingular elliptic curves have to be avoided in the previous process since they reduce the security of any cryptosystem based on the Di±e Hellman assumption on the elliptic logarithm. But more importantly, the process will be feasible whenever the probability to ¯nd a pair, (E; p), with a big prime factor qj jE(Fp)j is big enough. One problem arises naturally from the above.

Modularity of CM elliptic curves over division fields

Let E be a CM elliptic curve defined over an algebraic number field F. In general E will not be modular over F. In this paper, we determine extensions of F, contained in suitable division fields of E, over which E is modular. Under some weak assumptions on E, we construct a minimal subfield of division fields over which E is modular.

Anticyclotomic main conjectures for CM modular forms

Generalizing work of Agboola-Howard on CM elliptic curves, we prove a version of the main conjecture of Iwasawa theory over the anticyclotomic ℤp-extension of a quadratic imaginary field K for Selmer groups attached to modular forms with complex multiplication by K. We deduce a statement to the effect that if the L-function of a CM modular form f of even weight has odd order of vanishing at its central point, then a Selmer group attached to f over ℚ is infinite.

Computing the cardinality of CM elliptic curves using torsion points

Let ℰ/ℚ ¯ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field 𝕂. The field of definition of ℰ is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce ℰ modulo one the factors of p and get a curve E defined over 𝔽 p . The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose to use the action of the Frobenius on torsion points of small order built with class invariants generalizing the classical Weber functions.

Anticyclotomic Iwasawa theory of CM elliptic curves

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, the two variable main conjecture of Rubin implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety A A defined over a number field K K that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension K ( A [ p ∞ ] ) K(A[p^\infty ]) has no nonzero pseudo-null submodules.

The Ramanujan differential operator, a certain CM elliptic curve and Kummer congruences

Let τ be a point in the upper half-plane such that the elliptic curve corresponding to τ can be defined over Q ,a nd letf be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f ,w e obtain a family of modular forms D l f . In this paper we study the behavior of D l (f )(τ ) modulo the powers of a prime p> 3. We show that for p ≡ 1 mod 3 the quantities D l (f )(τ ), suitably normalized, satisfy Kummer-type congruences, and that for p ≡ 2m od 3 the p-adic valuations of D l (f )(τ ) grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if p ≡ 1 mod 3 and supersingular otherwise.

A formula for the central value of certain Hecke L-functions

Let N ≡ 1 mod 4 be the negative of a prime, K = Q ( N ) and O K its ring of integers. Let D be a prime ideal in O K of prime norm congruent to 3 mod 4 . Under these assumptions, there exists Hecke characters ψ D of K with conductor ( D ) and infinite type ( 1 , 0 ) . Their L-series L ( ψ D , s ) are associated to a CM elliptic curve A ( N , D ) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L ( ψ D , s ) of the form L ( ψ D , 1 ) = Ω ∑ [ A ] , I r ( D , [ A ] , I ) m [ A ] , I ( [ D ] ) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at | N | and infinity and [ A ] are class group representatives of K. An application of this formula for the case N = - 7 will allow us to prove the non-vanishing of a family of L-series of level 7 | D | over K.

Reduction of CM elliptic curves and modular function congruences

Here we investigate congruences for the coefficients of modular functions on SL2(ℤ). We obtain a simple criterion, in terms of the locus of supersingular j-invariants, which guarantees certain congruences. We implement this criterion in the case of modular functions obtained from Hilbert class polynomials.

Anticyclotomic Iwasawa theory of CM elliptic curves II

Anticyclotomic Iwasawa theory of CM elliptic curves II

On the field of definition for modularity of CM elliptic curves

The purpose of this paper is to decide the conditions under which a CM elliptic curve is modular over its field of definition.

Derivatives of Eisenstein series and Faltings heights

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = ½ of an Eisenstein series of weight for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = ½ is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = ½ coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish.

The main conjecture for CM elliptic curves at supersingular primes

At a prime of ordinary reduction, the Iwasawa for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi's conjecture relates modified Selmer groups, which he defined, with modified padic L-functions defined by the first author. In this paper we prove Kobayashi's conjecture for elliptic curves with complex multiplication.

Stark conjectures for CM elliptic curves over number fields

We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s=0. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.

Elliptic units in K2

We present certain norm-compatible systems in K 2 of function fields of some CM elliptic curves. We demonstrate that these systems have some properties similar to elliptic units.

Cyclicity of CM elliptic curves modulo 𝑝

Let E E be an elliptic curve defined over Q \mathbb {Q} and with complex multiplication. For a prime p p of good reduction, let E ¯ \overline {E} be the reduction of E E modulo p . p. We find the density of the primes p ≤ x p \leq x for which E ¯ ( F p ) \overline {E}(\mathbb {F}_p) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.

On the Tamagawa Number Conjecture for CM Elliptic Curves Defined Over ?

In this paper we prove, under the assumption that the Soulé regulator map is injective, that, for all integers k ⩾0, the description by the local Tamagawa number conjecture for CM elliptic curves defined over Q , corresponding to the values of their L -functions at k +2, is true.

On the Tamagawa Number Conjecture for CM Elliptic Curves Defined Over [formula omitted

In this paper we prove, under the assumption that the Soulé regulator map is injective, that, for all integers k⩾0, the description by the local Tamagawa number conjecture for CM elliptic curves defined over Q, corresponding to the values of their L-functions at k+2, is true.

On the p-adic realization of elliptic polylogarithms for CM-elliptic curves

Let E be a CM-elliptic curve over ℚ with good ordinary reduction at a prime p≥5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Beĭlinson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Grössencharakter associated to E.

L-series and their 2-adic valuations at s=1 attached to CM elliptic curves

$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given. Let $ E_1: y^{2}=x^{3}-D_1 x $ be elliptic curves over the Gaussian field $K=\Q(\sqrt{-1}), $ with $ D_1 =\pi_{1} ... \pi_{n} $ or $ D_1 =\pi_{1} ^{2}... \pi_{r} ^{2} \pi_{r+1} ... \pi_{n}$, where $\pi_{1}, ..., \pi_{n}$ are distinct primes in $K$. A formula for special values of Hecke $L-$series attached to such curves expressed by Weierstrass $\wp-$function are given; a lower bound of 2-adic valuations of these values of Hecke $L-$series as well as a criterion for reaching these bounds are obtained. Furthermore, let $ E_{2}: y^{2}=x^{3}-2^{4}3^{3}D_2^{2} $ be elliptic curves over the quadratic field $ \Q(\sqrt{-3}) $ with $ D_2 =\pi_{1} ... \pi_{n}, $ where $\pi_{1}, ..., \pi_{n}$ are distinct primes of $\Q(\sqrt{-3})$, similar results as above but for $3-adic$ valuation are also obtained. These results are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer, and develop some results in recent literature for more special case and for $2-adic$ valuation.