On the genera and spinor genera of integral, ternary quadratic forms
Tom 591 / 2024
Streszczenie
It is proved that two ternary forms $f$ and $g$ with the same elementary invariants, namely Sylvester signature $\varepsilon $, type $\mathbf{T}$, and $\Omega $ and $\Delta $, belong to the same genus if $aa_{1}$ is a square mod $\Omega $ and $ee_{1}$ is a square mod $ \Delta $, where $a$, $a_{1}$ (resp. $e$, $e_{1}$) are integers properly represented by $f$ and $g$ (resp. their reciprocals $\phi $ and $\gamma $) and coprime to $\Omega $ (resp. $\Delta $). This result depends on the existence of particular solutions of a Diophantine equation of the form $x^{2}+\Delta y^{2}=ez^{2}$. This provides a complete set of genus invariants, the $\tau $-symbols. A given putative $\tau $-signature is realizable by a form if and only if the $\tau $-symbols are linked together by a particular relation (the so called Great Relation). The $\tau $-signature is used to implement the Conway algorithm to calculate the number of spinor classes residing in a given genus, namely, the spinor kernel of a given genus is obtained directly from its $\tau $-signature. The spinor kernel is obtained from the so called group $p$-AN of $p$-adic automorph numbers of the genus, $p$ a prime number. The group $-1$-AN comes from $\varepsilon $. The group $p$-AN for $p \gt -1$ comes from the $\tau _{p}$-symbol. The case $p=2$ is complicated and a table is needed to present all possibilities, and this explains the difficulties found by earlier investigators to understand the so called Meyer Theorem. Using this table and the $\tau $-signature of a genus it is a quick task to obtain the number of spinor genera lying in it. Moreover, the knowledge of the spinor kernel is used to obtain representatives of all the spinor genera contained in a given genus. As an important corollary of these computations the problem raised by Meyer’s Theorem (see Jones (1950) and McCarthy (1957)) on indefinite forms is solved in all generality. The genera representing $j$ if $j=\pm 1,\pm 2$ or if $j$ is prime to $2\Omega $ are characterized and the number of spinor genera in a genus representing $j$ when $j=\pm 1,\pm 2$ or $j$ is prime to $2\Omega \Delta $ is obtained, and representatives of these spinor classes are constructed. When all classes in an indefinite genus represent such $j$’s, the construction of these classes is greatly facilitated by this fact, and this is used in the elaboration of a gallery of examples. The problem of ascertaining whether two given indefinite ternary forms in the same genus are integrally equivalent is solved.
