Affiliation: University of Wisconsin
Email: bringman@math.wisc.edu
Title Of Talk: On Fourier coefficients of Siegel cusp forms with small weight
Abstract: Let $f \in S_k(\Gamma_g)$ be a Siegel cusp form of degree $g$ and weight $k$ on $\Gamma_g$, the Siegel modular group. We want to prove the following \begin{satz} Let $g\geq 2$ be an integer and suppose that $k\geq g+1$; let $F \in S_k(\Gamma_g)$ with Fourier coefficients $a(T)$, where $T$ is a positive definite symmetric half-integral $g \times g$ matrix. Then we have \begin{displaymath} a(T)\ll_{\epsilon, F} (\det T)^{ k \slash 2 - 1 \slash (2g) -(1- 1 \slash g) \alpha_g + \epsilon} \qquad (\epsilon > 0), \end{displaymath} where $\alpha_g^{-1}:= 4(g-1) + 4 \left\lfloor \frac{g-1}{2} \right\rfloor+ \frac{2}{g+2}$ and where the constant implied in $\ll_{\epsilon,F}$ only depends on $\epsilon$ and $F$. \end{satz} For the proof one can use the Fourier-Jacobi decomposition of $F$ \begin{eqnarray*} F(Z)= \sum_{m>0} P_m(\tau,z)e^{2 \pi i \,\text{tr} ( m \tau')}, \end{eqnarray*} were the summation extends over all positive definite symmetric half-integral $(g-1) \times (g-1)$ matrices. Then the coefficients $ P_m(\tau,z)$ are Jacobi cusp forms. The case $k>g+1$ is treated by B\"ocherer and Kohnen. They use certain Poincar\'{e} series for the Jacobi group and develop a kind of Petersson coefficient formula. Unfortunately these series fail to converge absolutely in the case $k=g+1$. Therefore we use the so-called Hecke trick and multiply every summand of the Poincar\'{e} series with a factor depending on a complex variable $s$, such that the new series $P_{k,m;(n,r), s}$ is again absolutely convergent for $\sigma=$Re$(s)$ sufficiently large. Moreover this factor is chosen such that the new series is again invariant under the slash operation of the Jacobi group. Now the method is the following one: we compute the Fourier expansion of the Poincar\'{e} series $P_{k,m;(n,r), s}$, show that it is even absolutely and locally uniformly convergent in a larger domain of \mb{C}, that contains the point $s=0$ if $k=g+2$ and take it as a new definition for the Poincar\'{e} series in this larger domain. What is left to show is that these series are Jacobi cusp forms and that the Petersson coefficient formula is still valid. Afterwards we can show very easily the desired estimate.
Last update made Thu Nov 11 14:46:08 EST 2004.
Please report problems to:
frank@math.ufl.edu