On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials

For an imaginary quadratic fieldK we study the asymptotic behaviour (with respect top) of the number of integers inK with norm of the formk(p−k) for some 1≤k≤p−1, wherep is a prime number. The motivation for studying this problem is that it is known by recent results due to G. Frey and E. Kani that knowledge of this asymptotic behaviour can lead to statements of existence of curves of genus 2 with elliptic differentials in particular cases. We give a general, and from one point of view complete, answer to this question on asymptotic behaviour. This answer is derived from a theorem concerning the number of representations of a natural number by certain quaternary quadratic forms. This second result may be of some independent interest because it can be seen as a generalisation of the classical theorem of Jacobi on the number of representations of a natural number as a sum of 4 squares.