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Authors
Affiliations
1 Departement De Mathematiques, Ecole Polytechnique Federale De Lausanne, Lausanne, CH
Source
Journal of the Ramanujan Mathematical Society, Vol 26, No 2 (2011), Pagination: 219-235
Abstract
Let n be a square-free polynomial over 𝔽q, where q is an odd prime power. In this paper, we determine which irreducible polynomials p in 𝔽q [x] can be represented in the form X2+nY2 with X, Y ∈ 𝔽q [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 = 𝔽q (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.
