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On Eisenstein series in $M_{2k}(\Gamma_0(N))$ and their applications. (arXiv:1705.06032v1 [math.NT])
来源于:arXiv
Let $k,N \in \mathbb{N}$ with $N$ square-free and $k>1$. Let $f(z) \in
M_{2k}(\Gamma_0(N))$ be a modular form. We prove an orthogonal relation, and
use this to compute the coefficients of Eisenstein part of $f(z)$ in terms of
sum of divisors function. In particular, if $f(z) \in E_{2k}(\Gamma_0(N))$,
then the computation will to yield to an expression for Fourier coefficients of
$f(z)$. We give three applications of the results. First, we give formulas for
convolution sums of the divisor function to extend the result by Ramanujan.
Second, we give formulas for number of representations of integers by certain
infinite families of quadratic forms. And at last, we determine a formula for
Fourier coefficients of $f(z)\in E_{2k}(\Gamma_0(N))$, where $f(z)$ is an eta
quotient, and then we show that the set $\{ f(z) \in E_{2k}(\Gamma_0(N)), k
\geq 1 \}$ is finite for all $N \in \mathbb{N}$ square-free. 查看全文>>