# Siegel modular forms and Hecke operators in degree 2

JL Hafner, LH Walling

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

### Abstract

Let $F$ be a degree $2$ Siegel modular form of weight $k$, level $N$. $F$ has a Fourier development of the form $$F(Z)=\sum\sb{\Lambda, G} c(\Lambda) \exp(\pi i{\rm Tr}(^{\ssf T}GTG Z)),$$ where in the outer summation $\Lambda$ varies over all isometry classes of rank $2$ lattices associated with a positive semidefinite even integral quadratic form $Q$ with a matrix $T$ representing $Q$ on $\Lambda$ and, for each such $\Lambda$, the inner sum varies over a set of representatives $G\in O(\Lambda)\backslash {\rm GL}\sb{2}({\Bbb Z})$ ($O(\Lambda)$ is the orthogonal group of $\Lambda$). The authors refer to $c(\Lambda)$ as a lattice Fourier coefficient of $F$. They first give an explicit expression for the action of Hecke operators on $F$ and from this they deduce certain average relations between lattice Fourier coefficients and Hecke eigenvalues of $F$. These enable them to get a generating function for each family of average lattice Fourier coefficients (locally) in terms of the eigenvalues and a maximal lattice in the family. As a consequence, they derive a local-global property for these average lattice Fourier coefficients and a formula connecting local coefficients and the Satake parameters. They also find a factorization of the Koecher-Maass series of $F$ in terms of three zeta functions associated with $F$, namely, the spinor zeta function, the standard zeta function and a zeta function with Fourier coefficients depending on the eigenvalues and the average lattice Fourier coefficients for maximal lattices. Finally, they obtain optimal but relative bounds for the lattice Fourier coefficients provided the Ramanujan-Petersson conjecture holds. At the end, they raise several open questions related to this work.
Translated title of the contribution Siegel modular forms and Hecke operators in degree 2 English Millenial Conference on Number Theory, University of Illinois, Urbana-Champaign, IL, USA, 21-26 May 2000 MA Bennett, BC Berndt, N Boston, HG Diamond, AJ Hildebrand, W Philipp A K Peters 117 - 148 32 1568811462 Published - May 2002

### Bibliographical note

Conference Proceedings/Title of Journal: Number Theory for the Millennium II, Volume 2
Conference Organiser: Berndt, BC, Boston, N, Diamond, HG, Hildebrand, AJ & Philipp, W
Other identifier: 9781568811468

• ## Cite this

Hafner, JL., & Walling, LH. (2002). Siegel modular forms and Hecke operators in degree 2. In MA. Bennett, BC. Berndt, N. Boston, HG. Diamond, AJ. Hildebrand, & W. Philipp (Eds.), Millenial Conference on Number Theory, University of Illinois, Urbana-Champaign, IL, USA, 21-26 May 2000 (pp. 117 - 148). A K Peters.