Points of finite order on elliptic curves with complex multiplication

On Primitive Points of Elliptic Curves with Complex Multiplication

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 de...

Torsion 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...

Elliptic Curves and Complex Multiplication

On Fields of Definition of Torsion Points of Elliptic Curves with Complex Multiplication

For any elliptic curve E defined over the rationals with complex multiplication (CM) and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves; in particular, we classify all cases where there are torsion points over Galois number fields not containing the field of definiti...

A conjecture on the torsion points of elliptic curves with the complex multiplication

Let GA be an AF -algebra given by a periodic Bratteli diagram with the incidence matrix A. Depending on a polynomial p(x) ∈ Z[x], we assign to GA a finite abelian group Abp(x)(GA) = Z /p(A)Z. It is shown that for every p(x), such that p(0) = ±1, the Abp(x)(GA) is an invariant of the strong stable isomorphism class of the AF -algebra GA. Using a categorical correspondence between the elliptic cu...