A generalization of Watson transformation and representations of ternary quadratic forms

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• 作者：Jangwon Ju^a ; ^{jjw@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Inhwan Lee^a ; ^{lih0905@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Byeong-Kweon Oh^b ; ^{bkoh@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11E12 ; 11E20
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：167
• 期：Complete
• 页码：202-231
• 全文大小：529 K

文摘

Let L   be a positive definite (non-classic) ternary 51d8b44056b4b402d2961c5f9c0e5ac5" title="Click to view the MathML source">Z$\mathbb{Z}$-lattice and let p   be a prime such that a $\frac{1}{2}{\mathbb{Z}}_p$-modular component of L_p$L_p$ is nonzero isotropic and 4⋅dL$4\cdot dL$ is not divisible by p. For a nonnegative integer m  , let 1e21c44" title="Click to view the MathML source">G_L,p(m)${\mathcal{G}}_{L,p}(m)$ be the genus with discriminant p^m⋅dL$p^m\cdot dL$ on the quadratic space 1edd4" title="Click to view the MathML source">L^pm⊗Q$L^{p^m}\otimes \mathbb{Q}$ such that for each lattice 1e34832366c5ed15dfec1afc9c012" title="Click to view the MathML source">T∈G_L,p(m)$T\in {\mathcal{G}}_{L,p}(m)$, a $\frac{1}{2}{\mathbb{Z}}_p$-modular component of T_p$T_p$ is nonzero isotropic, and 51346436567faba7fcddad42c6824" title="Click to view the MathML source">T_q$T_q$ is isometric to (L^pm)_q${(L^{p^m})}_q$ for any prime q different from p  . Let r(n,M)$r(n,M)$ be the number of representations of an integer n   by a 51d8b44056b4b402d2961c5f9c0e5ac5" title="Click to view the MathML source">Z$\mathbb{Z}$-lattice M  . In this article, we show that if m⩽2$m⩽2$ and n is divisible by p   only when m=2$m=2$, then for any 1e34832366c5ed15dfec1afc9c012" title="Click to view the MathML source">T∈G_L,p(m)$T\in {\mathcal{G}}_{L,p}(m)$, r(n,T)$r(n,T)$ can be written as a linear summation of r(pn,S_i)$r(pn,S_i)$ and r(p³n,S_i)$r(p^{3}n,S_i)$ for S_i∈G_L,p(m+1)$S_i\in {\mathcal{G}}_{L,p}(m+1)$ with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute r(n,T)$r(n,T)$, for any 1e34832366c5ed15dfec1afc9c012" title="Click to view the MathML source">T∈G_L,p(m)$T\in {\mathcal{G}}_{L,p}(m)$, by using the number of representations of some integers by lattices in G_L,p(m+1)${\mathcal{G}}_{L,p}(m+1)$ for an arbitrary integer m.