# Themen der Hecke-Theorie zur orthogonalen Gruppe O(2,n+2)

• Topics of Hecke theory for the orthogonal group O(2,n+2)

Let $t\in \mathbb{N}$ be squarefree and $S_t:=\left(\begin{smallmatrix} 0&1 \\ 1&0 \end{smallmatrix} \right) \bot \left(\begin{smallmatrix} 0&1 \\ 1&0 \end{smallmatrix} \right) \bot (-2t)$. We consider $M_t(m):=\{M\in\mathbb{Z}^{5\times 5}; M^{tr}S_tM=m^2S_t\}$, $\Gamma_t:=M_t(1)$ and $\mathcal M_t:=\bigcup_{m\in \mathbb{N}}M_t(m)$. The Hecke algebra $\mathcal H:=\mathcal H (\Gamma_t,\mathcal M_t)$ is the tensor product of its $p$-primary components $\mathcal H_{p}:=\mathcal H(\Gamma_t,\bigcup_{k\in\mathbb{N}_0}M_t(p^k))$. These $p$-primary components are polynomial rings over $\mathbb{Z}$ in $\Gamma_t \operatorname{diag}(1,1,p,p^2,p^2) \Gamma_t,\;\;\Gamma_t \operatorname{diag}(1,p,p,p,p^2) \Gamma_t\;$ and $\;\Gamma_t \operatorname{diag}(p,p,p,p,p) \Gamma_t$, which are algebraically independent.It is well known that the orthogonal group is isomorphic to the maximal discrete extension $\Sigma_t$ of the paramodular group of degree two and level $t$. We transfer this result to the Hecke algebra $\widehat{\mathcal H}$ for $\Sigma_t$.Furthermore, $\Sigma_t$ is isomorphic to the discriminant kernel of $\Gamma_t$. The corresponding Hecke algebra is not commutative if $t>1$.More generally, we consider the orthogonal group $O(2,n+2)$, $n\in\mathbb{N}$, and describe a fundamental set of its operation on the upper half space. For $S$ Euclidian, i.e. such $S$ which admit a kind of Euclidian algorithm, we descibe a system of representatives for the right and double cosets of the similarity matrices. We can calculate the number of classes if $\det(S)=1$.The last chapter deals with applications on modular forms and Hecke algebras in the case that $S$ is Euclidian. We transfer known boundaries for cusp forms. The Eisenstein series are simultaneous Hecke eigenforms. If $\det(S)=1$, in particular for the $E_8$ lattice, we provide a commutation theorem for the orthogonal $\Phi$-operator and the $T(p)$-operator. As a result, we show that all Hecke eigenforms of the $T(p)$-operator with non-vanishing zeroth Fourier coefficient are already multiples of Eisenstein series. We complete the thesis with the result that in this case the $T(p)$-operators are self-adjoint.