Some results on congruence p-adic group actions on the (p+1)-regular tree, and on the representation of numbers in the form summation(i = 1 to 4) of C(i)(L(i)x(i) + a(i))(2).

Abstract

Fix integers $p>2$ and $L>1$ with p prime and $p\mid L.$ Let $Q\sp1(R)$ denote the group of standard quaternions of norm 1 over the ring R, and $P\sp1(R)=Q\sp1(R)/N(R),$ where $N(R)\le Q\sp1(R)$ is the central subgroup of scalar multiples of the identity. The action of an element of GL$\sb2(\doubq\sb{p})$ or $Q\sp1(\doubq\sb{p})$ on the $(p+1)$-regular tree ${\cal T}\sb{p}$ is characterized in terms of the element's trace and its determinant or norm. Some results are given on congruence subgroups $\Gamma\sb{p,L}$ and $\Gamma\sbsp{p,L}{\prime},$ defined as the kernel of the homomorphisms $Q\sp1(\doubz\lbrack1/p\rbrack)\to P\sp1(\doubz/(L))$ and $Q\sp1(\doubz\lbrack 1/p\rbrack)\to Q\sp1(\doubz/(L)).$ In particular, the number of vertices is found for the quotient graphs $\Gamma\sb{p,L}\\{\cal T}\sb{p}$ and $\Gamma\sbsp{p,L}{\prime}\\{\cal T}\sb{p}$ produced by the actions of these congruence groups on the $(p+1)$-regular tree. Several examples of these quotient graphs are generated using a computer program. Also, an asymptotic formula is given for the number of representations of an integer in the form $C\sb1y\sbsp{1}{2}+C\sb2y\sbsp{2}{2}+C\sb3y\sbsp{3}{2}+C\sb4y \sbsp{4}{2}$ subject to congruence conditions $y\sb{i}\equiv a\sb{i}$ (mod $L\sb{i}),$ where the $C\sb{i}$'s, $L\sb{i}$'s and $a\sb{i}$'s are integer constants, with the $C\sb{i}$'s and $L\sb{i}$'s positive. The formula is proved using Kloosterman's version of the circle method developed by Hardy. The asymptotic term is evaluated in the case of $C\sb{i}=1$ and $L\sb{i}=L,$ and necessary and nearly sufficient conditions are obtained for a positive integer to be representable.