A note on representation numbers of quadratic forms modulo prime powers
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Let $f$ be an integral quadratic form in $k$ variables, $F$ the Gram matrix corresponding to a $\Z$-basis of $\Z^{k}$. For $r\in F^{-1}\Z^{k}$, a rational number $n$ with $f(r)\equiv n\bmod\Z$ and a positive integer $c$, set $N_{f}(n,r;c):=\sharp\{x\in\Z^{k}/c\Z^{k}: f(x+r)\equiv n\bmod c\}$. Siegel showed that for each prime $p$, there is a number $w$ depending on $r$ and $n$ such that $N_{f}(n,r;p^{\nu+1})=p^{k-1}N_{f}(n,r;p^{\nu})$ holds for every integer $\nu>w$ and gave a rough estimation on the upper bound for such $w$. In this short note, we give a more explicit estimation on this bound than Siegel's.
Abstract
Let $f$ be an integral quadratic form in $k$ variables, $F$ the Gram matrix corresponding to a $\Z$-basis of $\Z^{k}$. For $r\in F^{-1}\Z^{k}$, a rational number $n$ with $f(r)\equiv n\bmod\Z$ and a positive integer $c$, set $N_{f}(n,r;c):=\sharp\{x\in\Z^{k}/c\Z^{k}: f(x+r)\equiv n\bmod c\}$. Siegel showed that for each prime $p$, there is a number $w$ depending on $r$ and $n$ such that $N_{f}(n,r;p^{\nu+1})=p^{k-1}N_{f}(n,r;p^{\nu})$ holds for every integer $\nu>w$ and gave a rough estimation on the upper bound for such $w$. In this short note, we give a more explicit estimation on this bound than Siegel's.
참고문헌 (5)
- A. Ajouz, "Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms", Ph. D., pp.2015
- B. C. Berndt, "Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts" (1998), John Wiley & Sons, Inc.
- J. H. Bruinier, "Eisenstein series attached to lattices and modular forms on orthogonal groups", Manuscripta Math, 106(4), pp.443~459 (2001)[DOI]
- J. H. Conway, "Sphere Packings, Lattices and Groups" (1999), Springer
- C. L. Siegel, "Uber die analytische Theorie der quadratischen Formen", Ann. of Math, 36(3), pp.527~606 (1935)[DOI]