Ternary Quadratic Forms And The Class Numbers Of Imaginary Quadratic Fields

Let n be an integer, and Q(x, y, z) a positive definite ternary quadratic form with integral coefficients. The representation number of n by Q is the number of integral solutions (x, y, z) of the equation Q(x,y, z) = n. We denote it by RQ(n). In this paper, we will study case by case for ternary quadratic forms: x2 + y2 + z2, x2 + y2 + 2z2, x2 + y2 + 3z2, x2 + y2 + 4z2, x2 + y2 + 5z2, x2 + y2 + 6z2, x2 + y2 + 8z2, x2 + 2y2 4- 3z2, x2 + 2y2 + 4z2, x2 + 2y2 + 6z2, x2 + 3y2 + 6z2, x2 + Ay2 + 4z2, x2 + 4y2 + 8z2, 2x2 + 2y2 + 3z2, 2x2 + 3y2 + 3z2, x2 + 2y2 + 2z2, x2 + 3y2 + 3z2,x2 + 5y2 + 5z2,x2 + 6y2 + 6z2,2x2 + 3y2 + 6z2,x2 + y2 + 2z2 + yz,x2+y2+7z2, x2+y2 +11z2, x2+y2 + 13z2, x2+y2 + 19z2, x2+3y2+5z2 and find the relations between their representation numbers and the class numbers of corresponding imaginary quadratic fields. Since the class numbers of ternary quadratic forms in last five cases are greater than 1, we need to associate with other representatives in the genus to establish the relations. We will formulate following results as examples.We assume that p is an odd prime. Let Q = x2 + y2 + 2z2 + yz. Then we have Similarly, let Q1 = x2 + 3y2 + 5z2, Q2 = x2 + 2y2 + 8z2 - 2yz. Then we have Where h(d) stands for the class numbers of Q((?)).We will give some "dual" results. For example, we haveThe result about ternary quadratic form x2 + y2 + 3z2 was first conjectured by Z.H.Sun and was verified by Guo-Peng-Qin in[3].In this paper,we will show that the obove phenomenon occurs in a much wider situation. We will first establish a series of similar relations for those diagonal positive ternary forms that the corresponding cusp form spaces are trivial and in which cases Pei has got the analytic formulas for the representation numbers. Moreover, we will study for those ternary quadratic forms whose corresponding cusp form spaces are not trivial and establish the analytic formulas and then get the similar relations as bove.For the ternary quadratic forms x2+py2+qz2, where p and q are odd primes, we obtain a formula which relates the representation numbers of these ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields.In the last chapter, we will verify one case of the conjectures raised by Cooper and Lam on the representation numbers of n2= x2+by2+cz2. More preciesely, we will prove that RQ(n2)= 4H(1,21,n), where and ep is the non-negative integer such that pep||n.We may verify more cases of the conjectures. For example, if b= 3, c= 10, we may verify it by similar way since the class number of the corresponding ternary quadratic form is equal to 1, and the basis of the corresponding space of Eisenstein series is known.