Compatibility of polynomial disk moments under refinement

For every prime natural number $p$, positive natural number $N$, natural number $k\ge 2$, and pair of unital ring homomorphisms $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, where $\overline{\mathbb Q}$ is an algebraic closure of $\mathbb Q$ and $\mathbb C_p$ is the completed algebraic closure of $\mathbb Q_p$, suppose the following data are given. There is a weight-$k$ cusp form $f$ for the image of $\Gamma_1(N)$ in $\mathrm{GL}_2(\mathbb R)$, a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$ (extended by zero on nonunits), and a sequence $c_m\in\overline{\mathbb Q}$ for $m\in\mathbb N$, such that the coefficient of degree $m$ in the period-one Fourier expansion of $f$ is $\iota(c_m)$, $c_1=1$, and, for every $\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\in\Gamma_0(N)$ and $z$ in the upper half-plane, $f((Az+B)/(Cz+D))=\iota(\varepsilon(D))(Cz+D)^k f(z)$. In addition, for every prime natural number $\ell$ and every such $z$, these data satisfy $\ell^{-1}\sum_{b=0}^{\ell-1}f((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}f(\ell z)=\iota(c_\ell)f(z)$. There are also two nonzero complex numbers $\Omega_s$, indexed by Boolean values $s$, and algebraic numbers $v_{s,t}(r)$ for every Boolean $s$, natural number $t$, and rational $r$, such that, whenever $t\le k-2$, $\iota(v_{s,t}(r))=(I_t(r)+\sigma_s(-1)^t I_t(-r))/(2\Omega_s)$, where $\sigma_{\mathrm{true}}=1$, $\sigma_{\mathrm{false}}=-1$, and $I_t(r)=2\pi\int_0^\infty f(r+iy)(r+iy)^t\,dy$ is the complex-valued Bochner integral with respect to real Lebesgue measure. The $\mathbb Z$-submodule of $\overline{\mathbb Q}$ generated by all $v_{s,t}(r)$ with $t\le k-2$ is assumed finitely generated; the values with $t>k-2$ are included in the data but have no comparison or finiteness requirement. Let $\alpha\in\mathbb C_p$ satisfy $\lVert\alpha\rVert=1$ and $\alpha^2-\iota_p(c_p)\alpha+\iota_p(\varepsilon(p))p^{k-1}=0$. For rational $a,m$ define $A_{s,j}(a,m)=\sum_{t=0}^{j}\binom jt m^t a^{j-t}v_{s,t}(-a/m)\in\overline{\mathbb Q}$, using the rational embeddings into $\overline{\mathbb Q}$, and, for an integer $a$ and a positive natural number $n$, define $D_{s,j}(n,a)=\alpha^{-n}\iota_p(A_{s,j}(a,p^n))-\iota_p(\varepsilon(p))p^{k-2}\alpha^{-(n+1)}\iota_p(A_{s,j}(a,p^{n-1}))$. Then for every Boolean $s$, every natural number $j\le k-2$, every positive natural number $n$, and every integer $a$, the identity $\sum_{b=0}^{p-1}D_{s,j}(n+1,a+bp^n)=D_{s,j}(n,a)$ holds in $\mathbb C_p$. The assertion includes $j=0$, $k=2$ (which forces $j=0$), $N=1$, $n=1$ (for which $p^{n-1}=1$), negative and zero $a$, and $a$ divisible by $p$; there is no coprimality assumption on $a$ or on $p,N$, nor any assumption that the Dirichlet character is primitive. The hypothesis $\lVert\alpha\rVert=1$ excludes $\alpha=0$, and $p\ge2$ and $n\ge1$ ensure that all rational denominators and exponents occurring in this identity are nondegenerate. The binomial sum includes its endpoints, with zeroth powers equal to one even when $a=0$; the theorem makes no assertion at depth $n=0$.