On the Gauss E ϒPHKA theorem and some allied inequalities

Abstract

We use the 1907 Hurwitz formula along with the Jacobi triple product identity to understand representation properties of two JP (Jones-Pall) forms of Kaplansky: 9x 2 + 16y 2 + 36z 2 + 16yz + 4xz + 8xy and 9x 2 + 17y 2 + 32z 2 − 8yz + 8xz + 6xy. We also discuss three nontrivial analogues of the Gauss E Υ PHKA theorem. The technique used can be applied to all known spinor regular ternary quadratic forms.

Exceptional sets for spinor regular ternary quadratic forms

“…For each of the 27 spinor regular ternaries that are alone in their spinor genus, Aygin et al [1] summarize in Table A.17 the list of all positive integers not represented by the form. As detailed in that paper, the results for several of these forms appeared earlier in work of Lomadze [12] and Berkovich [3]. However, for the remaining two forms which are not alone in their spinor genus, the method of [1] yields only the even integers that fail to be represented.…”

Spinor representations of positive definite ternary quadratic forms

“…The third and fourth assertions are trivial. Now, by using the above corollary and spinor representation theory of quadratic forms, we reprove Theorems (1.9), (1.10) and (1,11) in [3]. To do these, we define ternary quadratic forms:…”

“…We consider the second case which corresponds to Theorem (1.10) of [3]. Note that r s2 B2,w2 p8n`1, M 2 q " |tpx, y, zq t P Z 3 : x 2`y2`2 z 2 " 8n`1, px, y, zq " p1, 4, 0qpmod p4, 16, 2qqu| " |tpx, y, zq t P Z 3 : p4x`1q 2`p 16y`4q 2`2 p2zq 2 " 8n`1u| " |tpx, y, zq t P Z 3 : p4x`1q 2`1 6p4y`1q 2`8 z 2 " 8n`1u| " 1 4 |tpx, y, zq t P Z 3 : p2x`1q 2`1 6p2y`1q 2`8 z 2 " 8n`1u| " 1 4 |tpx, y, zq t P Z 3 : x 2`1 6y 2`8 z 2 " 8n`1,px, y, zq " p1, (iii) r s6 B6,w6 p16n`2, M 6 q " 0 if and only if 16n`2 " 2M 2 and all prime divisors of M are congruent to 1 modulo 4 and r s6 B6,w6 p16n`10, M 6 q ą 0.…”

“…If the corresponding subform has class number one or the spinor genus of the subform contains only one class, then we may give a formula on the number of representations of quadratic forms with some congruence conditions. By using this method, we reprove Theorems (1.9), (1,10) and (1,11) in [3], and prove some extensions of them. A (quadratic) Z-lattice L " Zx 1 `Zx 2 `¨¨¨`Zx n of rank n is a free Z-module equipped with a bilinear form B : L ˆL Ñ Z.…”