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Über dieses Buch

This book of tables includes a reduced representative of each class of. integral positive definite primitive quaternary quadratic forms through discriminant 1732. The classes are grouped into genera; also included are Hasse symbols, the number of automorphs and the level of each such form, and the mass of each genus. An appendix lists p-adic densities and p-adic Jordan splittings for each genus in the tables for p = 2 and for each odd prime p dividing the discriminant. The book is divided into several sections. The first, an introductory section, contains background material, an explanation of the techniques used to generate the information contained in the tables, a description of the format of the tables, some instructions for computer use, examples, and references. The next section contains a printed version of the tables through discriminant 500, included to allow the reader to peruse at least this much without the inconvenience of making his/her own hard copy via the computer. Because of their special interest, we include tables of discriminants 729 and 1729 at the end of this section. Limitations of space preclude publication of more than this in printed form. A printed appendix through discriminant 500 and for discriminants 729 and 1729 follows. The complete tables and appendix through discriminant 1732 are compressed onto the accompanying 3.5 inch disk, formatted for use in a PC-compatible computer and ready for research use particularly when uploaded to a mainframe. Documentation is included in the Introduction.



1. Introduction

The following computer-generated tables of reduced regular primitive positive definite quaternary quadratic forms over the rational integers are inspired by and are an outgrowth of the remarkable Brandt-Intrau tables of reduced positive ternary forms [2], published in pre-computer days after what must have been an incredible amount of effort. The Brandt-Intrau tables serve not only as a model of accuracy but also as a starting point for generating classes of reduced forms with an additional variable by computer. The technique used for this was essentially that of Germann [13], who began with the ternary forms through discriminant 16 from the Brandt-Intrau tables and computed all classes of reduced positive quaternary forms through discriminant 61. While Germann had few enough forms to allow separation into classes on an individual basis, our task was somewhat less manageable. Beginning with the ternary forms of discriminant δ ≤ 320 from the Brandt-Intrau tables (with the imprimitive ones adjoined), we generated a large file of quaternary forms containing representatives of all possible classes. Directed by a conjecture of John Hsia [14] that no two inequivalent such forms with the same discriminant would have the same theta series, two forms with the same discriminant were considered “equivalent” as a preliminary sorting measure if their first twenty theta coefficients were the same (the number 20 being chosen arbitrarily).
Gordon L. Nipp


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