Positive ternary quadratic forms with finitely many exceptions.

*(English)*Zbl 1129.11310The authors characterize the primitive integral positive definite ternary quadratic forms \(f\) which are “almost regular”, in the sense that the exceptional set \(E(f)\) consisting of those integers represented by the genus of \(f\), but not by \(f\) itself, is finite (integral quadratic forms for which \(E(f)\) is empty are referred to in the literature as “regular”). It is known that there are infinitely many equivalence classes of such forms \(f\), but only finitely many with \(E(f)\) of any prescribed size (for the particular case of regular forms, a complete list has been determined by W. C. Jagy, I. Kaplansky and A. Schiemann [Mathematika 44, No. 2, 332–341 (1997; Zbl 0923.11060)]). For a given positive integer \(k\), a procedure is described in this paper for obtaining an effective upper bound for the discriminant of a primitive integral positive definite ternary quadratic form \(f\) with \(|E(f)|\leq k\).

The characterization given for almost regular forms is best described in the language of quadratic lattices. For this purpose, let \(L\) be a primitive even integral \(\mathbb Z\)-lattice on a positive definite ternary quadratic space over \(\mathbb Q\). It is shown that \(L\) has exceptional set which is finite but nonempty if and only if (1) there is a unique prime \(p\) in the set \(\{2,3,5,7,11,13,17\}\) for which the local completion \(\mathbb Q_pL_p\) of the underlying space is anisotropic, (2) an associated lattice \(\lambda_{(p)}(L)\) has empty exceptional set, and (3) \(L\) represents all integers in an explicitly determined finite set \(P_L\) (which is empty unless there is a proper splitting of the genus of \(L\) into spinor genera). The lattice \(\lambda_{(p)}(L)\) is obtained from \(L\) by successive application of a transformation \(\lambda_{2p}\) which reduces the power of \(p\) dividing the discriminant but does not increase the size of the exceptional set.

The characterization given for almost regular forms is best described in the language of quadratic lattices. For this purpose, let \(L\) be a primitive even integral \(\mathbb Z\)-lattice on a positive definite ternary quadratic space over \(\mathbb Q\). It is shown that \(L\) has exceptional set which is finite but nonempty if and only if (1) there is a unique prime \(p\) in the set \(\{2,3,5,7,11,13,17\}\) for which the local completion \(\mathbb Q_pL_p\) of the underlying space is anisotropic, (2) an associated lattice \(\lambda_{(p)}(L)\) has empty exceptional set, and (3) \(L\) represents all integers in an explicitly determined finite set \(P_L\) (which is empty unless there is a proper splitting of the genus of \(L\) into spinor genera). The lattice \(\lambda_{(p)}(L)\) is obtained from \(L\) by successive application of a transformation \(\lambda_{2p}\) which reduces the power of \(p\) dividing the discriminant but does not increase the size of the exceptional set.

Reviewer: Andrew G. Earnest (Carbondale)