In 1927, Artin hypothesized that for a given non-zero integer other than 1, -1 or a perfect square, the density of primes p for which a generates the multiplicative group (Z/pZ)^* is positive. This problem is still open today. In 1976, Lang and Trotter formulated the elliptic curve analogue of Artin's primitive root conjecture. As the first step towards this conjecture, Serre investigated how often the reduction of an elliptic curve defined over the rational numbers modulo primes is cyclic and conjectured that it has a positive density. The Drinfeld module is the function field analogue of elliptic curves. In this talk, we will discuss Artin's conjecture and Serre's cyclicity conjecture in Drinfeld's modules. This is joint work with David Tweedle.

 

Organizer: Chun-Yen Shun (NTU)