Extra twists of Siegel modular forms, the associated central simple algebras and dimensions of spaces of Siegel modular forms

Abstract

Denote the Siegel upper half space as $$\mathbb{H}_2=\{X+iY \in M_2(\mathbb{C})~|~ X=X^t, Y=Y^t,Y>0\}. $$

A holomorphic function $F:\mathbb{H}_2 \rightarrow \mathbb{C}$ is called a Siegel modular form of weight $k$, if it satisfies certain conditions. Associated to a Siegel modular form we have the associated Galois representation : \[ \rho_{F, \lambda}: G_{\Q} \rightarrow \mathrm{GSp}_{4}(\overline{K_{\lambda}}). \]

Assume $\mathfrak{g}_l$ to be the lie algebra of the image of the representation and \[ \mathfrak{a}_l = \{u\in \mathfrak{gsp}_{4}(K \otimes_{\Q} \Q_l)=\prod_{\lambda \mid l}\mathfrak{gsp}_{4}(K_{\lambda})~| ~\Lambda(u) \in \Q_l \} \] where $\Lambda$ denotes the symplectic similitude.

Let $\Gamma=\mathrm{Hom}(K, \C^{\times})$. Then a Siegel modular form $F$ is said to have an {\it extra twist} if there exists a tuple $(\gamma, \chi_{\gamma})$ with $\gamma \in \Gamma$, and an associated $\chi_\gamma$ such that $\gamma(\rho_F) \cong \rho_F \otimes \chi_\gamma$.

One of the results if of the thesis shows that $\mathfrak{g}_l \subsetneq \mathfrak{a}_l$ if and only if $F$ admits an extra twist.

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While for a form with a non trivial character $\epsilon$, the pair $(c,\epsilon^{-1})$ is already an extra twist using Petersson inner product, we show examples in this thesis of how different ways of lifting classical modular forms with twists to Siegel modular forms gives rise to extra twists or not in the most natural sense. The lifts we cover include Yoshida lifts, Saito Kurokawa lifts and $\mathrm{Sym}^3$ lifts.

Assume $X$ to be the cross product algebra associated with the cocycle $c(\gamma, \delta)$ that is defined by $$ X = {\bigoplus}_{\gamma \in \Gamma} \> K \cdot x_{\gamma} $$

We introduce the intertwining opertator as well which is described as follows :

For a subgroup $U \subset G_{\Q}$ and a vector space $V$ over $K_{\lambda}$, let us denote by \[ \iota_{\lambda}^U(V):=\{ \Phi: (V, \Psi) \rightarrow (V, \Psi), \Phi \circ \g_{\lambda}(h)= \g _{\lambda}(h) \circ \Phi \text { for all } h \in \log(U)\}. \]

Then, in one of our second major results we show that $$\iota_l^{\overline{\mathfrak{g}_l}}(\overline{V_l})\simeq X \otimes_{\Q}\overline{\Q_l}.$$

We also share some results on the local Brauer classes of the central simple algebra $X$.

Due to Poor and Yuen, we have the restriction map $\phi_s^* : S_2^k(\Gamma_0^2(p)) \rightarrow S_1^{2k}(\Gamma_0(p))$. Since the right hand side of this equation is the space of classical modular forms, we derive information of Siegel modular forms using this map.

We have devised a program that gives the image of the restriction map of a Siegel modular form under arbitrary prime level.

We further compute an upper bound for the dimension of the space of Siegel modular form for a prime power.