The Lang-Trotter conjecture for Drinfeld modules - Tweedle, David (University of Waterloo, 2011)
In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.
Keywords: Lang-Trotter conjecture; Drinfeld modules; Artin's conjecture; Function fields; Pure Mathematics