Torsion in Rank 1 Drinfeld Modules and the Uniform Boundedness Conjecture

It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld A-module of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an L-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over L which have nonzero torsion. For the case A = Fq[T ], r = 1, and L = Fq(T ), we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the p-primary part of the torsion for rank 2 Drinfeld Fq[T ]-modules over a fixed function field. 1. Conjectures and Theorems In a 1977 paper, Mazur [13] proved that if E is an elliptic curve over Q, its torsion subgroup is one of the following fifteen groups: Z/NZ, 1 ≤ N ≤ 10 or N = 12; Z/2Z × Z/2NZ, 1 ≤ N ≤ 4. In particular, there is a uniform bound for the size of the torsion subgroup of an elliptic curve over Q. (Although this in recent years has often been called Ogg’s conjecture, it was essentially formulated by Levi [11] in 1908, and was again formulated by Nagell [15] in 1949, before the appearance of Ogg’s paper [16] in 1971. See [19] for more history.) Recently, this has been generalized to arbitrary number fields. Merel [14] proved that for every d ≥ 4, the order of the torsion subgroup of an elliptic curve E over a number field K of degree d over Q is divisible only by primes less than 2(d!). Earlier results of Kamienny and Mazur [10] and Abramovich [1] established bounds for d ≤ 14. Also, Kamienny and Mazur [10] proved that the exponent of each such prime is bounded, with the bound depending only on d. (Earlier work of Manin [12] proved the boundedness of each exponent for fixed K.) Together, these results imply that there is a bound N(d) depending only on d such that the order of the torsion subgroup of an elliptic curve over a number field of degree d over Q is at most N(d). This paper considers the analogous questions for Drinfeld modules. Date: June 19, 1995. 1991 Mathematics Subject Classification. Primary 11G09.