Representations of squares by septenary quadratic forms and Shimura lifts
Graduate project, Sun Yat-sen University, 2023
H. Chen, D. Ye and M. Yuan, On a remark of Cooper, Lam and Ye, Colloquium Mathematicum, 171 (2023), no.1, 127–144.
Project description
In this project, we made use of modular forms and the Shimura lifts to formulate the number \(r_{(a_{1},\ldots,a_{7})}(n^{2})\) of integral solutions of
\[a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}+a_{4}x_{4}^{2}+a_{5}x_{5}^{2}+a_{6}x_{6}^{2}+a_{7}x_{7}^{2}=n^{2}\]for a positive integer \(n\) and some given septenary tuple \((a_{1},\ldots,a_{7})\). For example, we obtained for \(n=\prod_{p\geq2}p^{e_{p}},\)
\[r_{(1,1,1,1,1,2,4)}(n^{2})= \begin{cases}6s(n)+4k(n)&\mbox{if $e_{2}=0$,}\\\\ 192s(n)-20g(n)&\mbox{if $e_{2}=1$,}\\\\ \frac{1491\times 2^{5e_{2}-3}-756}{2^{5}-1}s(n)&\mbox{if $e_{2}\geq2$,}\end{cases}\]where
\(s(n)=\prod_{p\geq3}\left(\frac{p^{5e_p+5}-1}{p^5-1}-p^2\left( \frac{-2}{p}\right)\frac{p^{5e_p}-1}{p^5-1}\right),\) and \(g(n)=\prod_{p\geq3}\left(A(p^{e_p})-p^2\left( \frac{-2}{p}\right)A(p^{e_p-1})\right)\)
with
\[\sum_{n=1}^{\infty}A(n)q^n =q\prod_{j=1}^{\infty}(1-q^{2j})^{12}.\]