Levels of Positive Definite Ternary Quadratic Forms

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character Xg ■ We define ä collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given level. We also give a formula for the number of genera of ternary forms with a given level and character. Introduction In this article, we consider some questions concerning the classification of positive definite ternary quadratic forms. Our motivation is the connection between quadratic forms and modular forms which is given in the theorem below. We first recall some notation and terminology concerning modular forms. Define a symbol (a/b) for a, b e Z by the following conditions: (1) (a/b) is the Legendre symbol if b is an odd prime. (2) (a/2) = (_i)(«2-D/8 if a is odd. (3) (fl/-l) = 1 if a>0, (a/-l) = -l if a<0. (4) ia/b) = 0 if gcd(a, b) > I, (1/0) = 1, (a/0) = 0 if a # 1. (5) (a/bc) = (a/b) • (a/c) for all b,ceZ. If t is a nonzero integer, define a function Xt on the integers as follows: Let t = qr2 with q squarefree. If q = 1 (mod 4), let D = q. If q = 2,3 (mod 4), let D = 4q . Then Xtin) = iD/n) for all n e Z. The function Xt is a quadratic Dirichlet character with conductor \D\ [11]. Let k be an integer, TV a positive integer (divisible by 4 if k is odd), and X a character modulo N. Let YoiN) be the subgroup of SL2(Z) consisting of all \abd\ with c = 0 (mod N). A modular form 6 is said to have weight k/2, level N, and character x if for all y = [acbd] e r0(7V) and all z e C with lm(z)>0, faz + b\ = f X(d) • (cz + d)k'2 6iz) if k is even, \cz + d) ~ \ xid) j\y, z)k 0(z) if k is odd. Here, Hy, z) e^xxdd)icz + d)l/2, where e¿ = 1 or i as d = 1 or 3 (mod 4). Denote the vector space of all such modular forms as Mk/2iN, x), Received July 24, 1990; revised February 25, 1991. 1991 Mathematics Subject Classification. Primary 11E20; Secondary 11E45. © 1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page