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Positive ternary quadratic forms with finitely many exceptions. (English) Zbl 1129.11310
The 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.

##### MSC:
 1.1e+21 General ternary and quaternary quadratic forms; forms of more than two variables 1.1e+13 Quadratic forms over global rings and fields
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