Tetsuo Nakamura

Cyclic torsion of elliptic curves

Let E be an elliptic curve over a number field k such that Endk E = Z and let w(k) denote the number of roots of unity in k. Ross proposed a question: Is E isogenous over k to an elliptic curve E′/k such that E′(k)tors is cyclic of order dividing w(k)? A counter-example of this question is given. We show that E is isogenous to E′/k such that E′(k)tors ⊂ Z/w(k)2Z. In case E has complex multiplication and Endk E = Z, we obtain certain criteria whether or not E is isogenous to E′/k such that E′(k)tors ⊂ Z/2Z.

On two-dimensional formal groups over the prime field of characteristic p > 0

In what follows, by a formal group we always mean a commutative one. In Hill [6], the following result (Theorem E’) is proved: Let F and G be onedimensional formal groups over F,,, the prime field of characteristic p > 0. If F and G are of finite height and have the same characteristic polynomial, then F and G are isomorphic over F,. For two-dimensional formal groups over F,, this is not valid in general. A counterexample is given in Section 4. Using a result on a classification of two-dimensional formal groups due to Ditters [4], we shall show in Section 2 that for certain types of formal groups, the above result is also valid (Corollary 1 of Theorem 1). In Section 3, we shall determine the simple twodimensional formal groups over F, whose endomorphism rings are maximal orders. Some examples are discussed in Section 4.

Elliptic ℚ-Curves with Complex Multiplication

Let H be the Hilbert class field of an imaginary quadratic field K. An elliptic curve E over H with complex multiplication by K is called a ℚ-curve if E is isogenous over H to all its Galois conjugates. We classify ℚ-curves over H, relating them with the cohomology group H2(H/ℚ, +1). The structures of the abelian varieties over ℚ obtained from ℚ-curves by restriction of scalars are investigated.

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .