Quaternionic modular forms of degree two over Q(-3,-1)

TL;DRAbstract

This thesis mainly deals with the analysis of quaternionic modular forms of degree two over the maximal order $O=+(1+i_1sqrt{3})/2+ i_2+(i_2+i_1i_2sqrt{3})/2$, where $H=R+i_1R+i_2R+i_1i_2R$ is the skew field of quaternions. These are defined as follows: Let $Sp_2(O)leqGL_4(O)$ be the quaternionic modular group over $O$ and $mathcal{H}(H)subsetH^{2imes 2}otimes_{H}C$ the quaternionic half-space of degree two, $GammaleqSp_2(O)$ a subgroup and $u$ a multiplier system of weight $k$ -- note that all these terms are defined in the thesis in detail. Then a holomorphic function $f:mathcal{H}(H)ightarrowC$ is called a quaternionic modular form of degree two with respect to $Gamma$ and $u$ of weight $k$ if [f((AZ+B)(CZ+D)^{-1})=u(M)(det(check{C}check{Z}+check{D}))^{k/2} f(Z)] holds for all $M=igl(egin{smallmatrix} A & B \ C & D end{smallmatrix}igr)inGamma$, with $~^{vee};$'' denoting the standard embedding of $H$ in $C^{2imes2}$. One of the main issues is to determine the graded ring of all quat

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