Eisenstein series – these are relatively well-known. Finally the statement and proof of the main theorem, and the subsequent existence of the $p$-adic $L$-function, are given in Section \[interpolation\].

Siegel modular forms {#modformsection}
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This section runs through the very basics of the modular forms that we study and their Fourier expansions are detailed.

For any matrix $A\in M_n(\mathbb{C})$, note the use of the following notation: $A>0$ ($A\geq 0$) to mean that $A$ is positive definite (resp. positive semi-definite), $|A| := \det(A)$, $\|A\| :=|\det(A)|$, and $\widetilde{A} := (A^T)^{-1}$. For any collection $A_1, \dots, A_r$ of matrices, let ${\operatorname{diag}}[A_1, \dots, A_r]$ be the matrix whose $j$th diagonal block is $A_j$ and is zero off the diagonal.

Let ${\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ and $\mathbb{I}_{\mathbb{Q}}$ denote the adele ring and idele group, respectively, of $\mathbb{Q}$. The Archimedean place is denoted by $\infty$ and the non-Archimedean places by $\mathbf{f}$. Recall the adelic norm $$|x|_{\mathbb{A}} = \prod_v|x_v|_v$$ where $x\in{\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ and each valuation, $|\cdot|_v$, is normalised. Let $\mathbb{T}$ denote the unit circle and define three $\mathbb{T}$-valued characters on $\mathbb{C}$, $\mathbb{Q}_p$, and ${\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ respectively by $$\begin{aligned}
    e:z&\mapsto e^{2\pi iz}, \\
    e_p:x&\mapsto e(-\{x\}), \\
    e_{\mathbb{A}}:x&\mapsto e(x_{\infty})\prod_p e_p(x_p),\end{aligned}$$ where $\{x\}$ denotes the fractional part of $x\in\mathbb{Q}_p$. If $x\in {\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ then we also 