6 Quaternionic Modular Forms

6.1 Definition

We define quaternionic modular forms over definite quaternion algebras explicitely as the following space.

Definition 6.1

Let \(\kappa : \mathbb {Z}_p^\times \to \mathcal{O}_p^\times \) be a locally analytic character. Given \(\alpha \in \mathbb {N}\), let

\[ \Sigma _\alpha = \left\{ \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{M}_2(\mathbb {Z}_p) : p^\alpha \mid c, \; p \nmid d, \; \det (\gamma ) \neq 0 \right\} . \]

The weight \(\kappa \) action of \(\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Sigma _\alpha \) on the Tate algebra \(\mathcal{A}_p\) with coefficients in \(\mathbb {C}_p\) is given by the continuous \(\mathbb {C}_p\)-linear extension of the map sending

\[ z^k \mapsto \frac{\kappa (c z + d)}{(cz + d)^2} \left( \frac{az+b}{cz+d} \right)^k \]

and given \(f(z) \in \mathcal{A}_p\) we write \((f||_\kappa \gamma )(z)\) for this action.

Fix \(\alpha \) and \(\kappa \) in the above, let \(D\) be a definite quaternion algebra and let \(U\) be an open compact subgroup of \(D_f^\times \), where \(D_f = D \otimes _{\mathbb {Q}} \mathbb {A}_f\) is \(D\) over the finite adeles, of wild level \(\geq p^\alpha \); that is the projection \(U \to D_{p}^\times \) is contained in \(\Sigma _\alpha .\) Now let \(A\) be any right \(\Sigma _\alpha \)-module.

Then the level \(U\), weight \(\kappa \) space of automorphic forms is the space

\[ \mathcal{L}(U,A) = \{ \varphi : D_f^\times \to A \; | \; \varphi (dgu) = \varphi (g) ||_\kappa u_p, \; \forall d \in D^\times , \; g \in D^\times _f, \; u \in U \} . \]

6.2 Finite Dimensional

Lemma 6.2

Using the above notation and supposing \(D_f^\times = \coprod _{i \in I} D^\times c_i U\) is the decomposition into finite double cosets, then

\[ \mathcal{L}(U,A) \cong \bigoplus _{i \in I} A^{\Gamma _i} \]

as a \(\mathbb {C}_p\) vector space, where \(\Gamma _i = c_i^{-1} D^\times c_i \cap U\).

6.3 Hecke Operators

We can define Hecke operators on these automorphic forms. We will keep the notation from the above definitions. Given \(\varphi \in \mathcal{L}(U,A)\) we define a right action of \(U\) on \(\mathcal{L}(U,A)\):

\[ (\varphi |_\kappa u) (g) := \varphi (gu^{-1}) ||_\kappa u_p. \]

With this we have:

Definition 6.3

Let \(\nu \in D^\times _f\) such that \(\nu _p \in \Sigma _\alpha \). Then the double coset \(U\nu U\) may be written as a disjoint union

\[ U \nu U = \coprod _{t \in T} U \nu _t \]

where \(T\) is a finite set, and \(\nu _t \in D_f^\times \). The Hecke operator is the map \([U \nu U] : \mathcal{L}(U,A) \to \mathcal{L}(U,A)\) given by

\[ [U \nu U] \varphi := \sum _{t \in T} \varphi |_\kappa \nu _t. \]

Definition 6.4

We set \(\varpi _l\) to be the element of \(\mathbb {A}_f\) which is \(l\) at \(l\) and 1 at all other places, and set \(\eta _l = \begin{pmatrix} \varpi _l & 0 \\ 0 & 1 \end{pmatrix}\).
The standard Hecke operators are then given by

\[ T_l := [U \eta _l U] \quad \text{and} \quad S_l:=[U \varpi U] \]

and if \(l = p\), we write \(T_p = U_p\).