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 field K we study the asymptotic behaviour ( with respect to p) of the number of integers in K with norm of the for m k(p - k) for some 1 less than or equal to k less than or equal to p - 1, where p is a prime number. The motivation for studying this probl em is that it is known by recent results due to G. Frey and E. Kani th at knowledge of this asymptotic behaviour can lead to statements of ex istence 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 numbe r by certain quaternary quadratic forms. This second result may be of some independent interest because it can be seen as a generalisation o f the classical theorem of Jacobi on the number of representations of a natural number as a sum of 4 squares.