Primes of the form X² + nY² in function fields
Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 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,sqrt(-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 X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for 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.
weak representation, strong representation, principal form, normalizing field, j-invariant
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