Representation numbers by quadratic forms of certain levels
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Let $Q$ be an integral positive definite quadratic form of level $N$ in $2k(\geq4)$ variables. We assume that $(-1)^kN$ is a fundamental discriminant and the associated character $\chi$ of $Q$ is primitive of conductor $N$. Under our assumption, we find the pairs $(k,N)$ such that the dimension of spaces of cusp forms of weight $k$ and level $N$ with Nebentypus $\chi$ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs $(k,N)$, we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level $N$ in $2k$ variables, which are expressed in terms of divisor functions and their convolution sums.
Abstract
Let $Q$ be an integral positive definite quadratic form of level $N$ in $2k(\geq4)$ variables. We assume that $(-1)^kN$ is a fundamental discriminant and the associated character $\chi$ of $Q$ is primitive of conductor $N$. Under our assumption, we find the pairs $(k,N)$ such that the dimension of spaces of cusp forms of weight $k$ and level $N$ with Nebentypus $\chi$ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs $(k,N)$, we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level $N$ in $2k$ variables, which are expressed in terms of divisor functions and their convolution sums.
참고문헌 (14)
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