On Primitive Points of Elliptic Curves with Complex Multiplication
نویسندگان
چکیده
Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set ME,P explicitly.
منابع مشابه
Generating Elliptic Curves over Finite Fields Part I: Generating by Complex Multiplication
We study the theory of rational points on elliptic curves over nite elds and the theory of complex multiplication through which we construct elliptic curves over F p such that their orders of the group of rational points over F p are of the form mr where r is a prime and m is a small integer.
متن کاملTorsion Points on Elliptic Curves with Complex Multiplication
i.e., the supremum of all orders of torsion points on elliptic curves defined over some degree d number field. Write T (d)′ for the set of prime divisors of elements of Td, and P (d) for the largest element of T (d)′. Let TCM(d) (resp. TIM(d)) be the subset of T (d) corresponding to elliptic curves with complex multiplication (resp. with algebraic integral modulus j(E)), and similarly adding th...
متن کاملEfficient elliptic curve cryptosystems
Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...
متن کاملPoint Counting on Reductions of Cm Elliptic Curves
We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM Q-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.
متن کاملOn the Tate-shafarevich Groups of Certain Elliptic Curves
The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.
متن کامل