Division fields of \(Y^ 2=X^ 3-aX\)
Consider an elliptic curve E with j-invariant \(j=1728\) over \({\mathbb{Q}}\) which has a \({\mathbb{Q}}\)-rational point P of infinite order. For each prime p where the curve has good reduction, the point P defines a cyclic subgroup of the group \(E({\mathbb{F}}_ p)\) of \({\mathbb{F}}_ p\)-rational points on the curve. One can ask how often it actually generates this group. The question is a variant of similar ones in the setting of a unit of infnite order in an algebraic number field. Of course, in certain cases we know a priori that we don't obtain a generator. Whenever \(E({\mathbb{F}}_ p)\) is not cyclic we will not. Also, if the point P is an image \(\phi\) (Q) of some other rational point under an endomorphism \(\phi\) such that \(\phi\) mod p is defined over \({\mathbb{F}}_ p\) and doesn't define an isomorphism on \(E({\mathbb{F}}_ p)\) we also don't have a generator. The author exhibits two examples where the above obstructions happen for all primes \(p\equiv 1 mod 4\). He proves under the generalized Riemann hypothesis (GRH), that except in these two cases the set of primes \(p\equiv 1 mod 4\) for which P gives a generator, has a positive density. In earlier work he already derived (using GRH) a formula for this density. It involved the degrees over \({\mathbb{Q}}(i)\) of compositions of fields \({\mathbb{Q}}(i,\alpha^{-1}\bar O,\beta^{-1}P)\), for \(\alpha\),\(\beta\in End(E)\cong {\mathbb{Z}}[i]\). The current paper gives explicit formulas for the degree of these fields. This enables the author to derive his density results.
generator of set of rational points on a curve
scientific article; zbMATH DE number 4179523