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Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. (English) Zbl 0692.10020

The authors make use of recent advances in the theory of modular forms of half-integral weight and the arithmetic theory of quadratic forms to obtain a result which could be proved so far only in more than three variables: For a positive definite ternary integral quadratic form q the integral points on the ellipsoid \(q(x)=n\) are uniformly distributed for \(n\to \infty\). Moreover, an asymptotic formula is given for the number of points \(x\in {\mathbb{Q}}^ 3\) in a region of \(q(x)=n\) which satisfy some fixed congruence condition modulo \({\mathbb{Z}}^ 3.\)
The first part of the paper reduces the problem to the formula for the number of points on the whole ellipsoid. It is based on results by H. Iwaniec [Invent. Math. 87, 385-401 (1987; Zbl 0606.10017)] and the first-named author [Invent. Math. 92, 73-90 (1988; Zbl 0628.10029)]. A similar result for the sphere, but with a worse error term, had been obtained by E. P. Golubeva and O. M. Fomenko [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 160, 54-71 (1987; Zbl 0634.10043)]. The second part then deals with the representation number problem. The steps of the solution come all from the literature, in particular from papers of the second-named author [Invent. Math. 75, 283- 299 (1984; Zbl 0533.10021); J. Reine Angew. Math. 352, 114-132 (1984; Zbl 0533.10016); Nagoya Math. J. 102, 117-126 (1986; Zbl 0566.10015)].
Reviewer: H.-G.Quebbemann

MSC:

11E12 Quadratic forms over global rings and fields
11E16 General binary quadratic forms
11P21 Lattice points in specified regions
11D85 Representation problems
11F11 Holomorphic modular forms of integral weight
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References:

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