Primitive Points on Elliptic Curves [chapter]

S. Lang, H. Trotter
2000 Collected Papers Volume II  
A well-known conjecture of Artin predicts the density of primes for which a given rational number is a primitive root (cf. the introduction to his collected works). Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. We ask for the density of those primes p such that the group A(¥ p ) of rational points mod p is cyclic, generated by the reduction a of a mod p. We shall use
more » ... Galois extensions K x = Q04;, l~la) analogous to the splitting fields of the equations X 1 -a = 0 when a is in the multiplicative group. We may say that a is primitive for such primes. We let (a ) be the cyclic group generated by a. The affine group, equal to the extension of the translation group A x by GL 2 (l), operates on I' 1 a. For simplicity we fix an element w 0 G I"" 1 a. Then we may represent an element o in the affine group by a pair (7, r) with 7 € GL 2 (J) and a translation T EA l9 such that (7, T)U = u 0 + 7(w -u 0 ) + r. The Galois group Gal^/QC^/)) can be identified with a group of translations, subgroup of A l9 and is equal to A x for almost all / by a theorem of Bashmakov [Ba]. If o = (7, r) as above, we have ou = u if and only if (7 -l)(w 0 -u) = r. Let A be the discriminant of the curve. We want to give a condition on the Frobenius element a p = (y p9 r p ) in G t when p\ M in order that the index of {a > in ^4(F ) is divisible by /. Note that / divides the order of -4(F) if and only if 7 p has eigenvalue 1. Furthermore, A(F ) = Ker (7 -1). If y p = 1 then the index of (a) is divisible by /. Suppose on the other hand that Ker(7 p -1) is cyclic of order /. Then the index of (a ) is divisible by / if and only if there exists b G Â with lb = a and b is fixed by a p . Indeed, if a has period divisible by /, and the index is divisible by /, then a is divisible by / in ^4(F p ), otherwise Â(f p ) would contain Z(f) 2 . The converse is clear. If a has period not divisible by / then lb = a for some b in G>, so the assertion is also clear in this case. We see that the index of {a) is divisible by I if and only if o p lies in the AMS (MOS) subject classifications (1970). Primary 12A75, 14G25. iBoth authors supported by NSF grants.
doi:10.1007/978-1-4612-2120-3_19 fatcat:zjjyj5hz5nbh5ccb6cd25r7mbq