On the representation numbers of ternary quadratic forms and modular forms of weight 3/2

In this paper, we give an alternative proof of Berkovich and Jagy's genus identity and verify a series of conjectures raised by Cooper and Lam on the number of solutions of n2=x2+by2+cz2n2=x2+by2+cz2 and two conjectures raised by Sun on the number of solutions of p=x2+y2+3z2p=x2+y2+3z2, 3p=x2+y2+3z23p=x2+y2+3z2.