Primes of the Form X2+nY2 in Function Fields and Drinfeld Modules

Let n be a square-free polynomial over 𝔽_q, where q is an odd prime power. In this paper, we determine which irreducible polynomials p in 𝔽_q [x] can be represented in the form X²+nY² with X, Y ∈ 𝔽_q [x]. We restrict ourselves to the case where X²+nY² is anisotropic at ∞. As in the classical case over ℤ discussed in [2], the representability of p by the quadratic form X²+nY² is governed by conditions coming from class field theory. A necessary (and almost sufficient) condition is that the ideal generated by p splits completely in the Hilbert class field H of K = 𝔽_q (x, √ -n) (for the appropriate notion of Hilbert class field in this context). In order to get explicit conditions for p to be of the form X²+nY², we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial of H/K. This algorithm generalizes to all situations an algorithm of D.S. Dummit and D. Hayes for the case where -n is monic of odd degree.