Existence of the ordinary p-adic L-measure with MTT interpolation

For every prime natural number $p$, every positive natural number $N$, every natural number $k\ge 2$, every 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 every datum $f$ consisting of a weight-$k$ cusp form $F$ on the image of $\Gamma_1(N)$ in $\mathrm{GL}_2(\mathbb R)$, a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $a:\mathbb N\to\overline{\mathbb Q}$ such that the coefficient of degree $n$ of the width-one $q$-expansion of $F$ equals $\iota(a_n)$ for every $n\ge0$, $a_1=1$, $F(\gamma z)=\iota(\varepsilon(d))(cz+d)^kF(z)$ for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and every $z$ in the upper half-plane, and $\ell^{-1}\sum_{b=0}^{\ell-1}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}F(\ell z)=\iota(a_\ell)F(z)$ for every prime natural number $\ell$ and every such $z$, the hypothesis $\lVert\iota_p(a_p)\rVert=1$ implies the existence of $\alpha\in\mathbb C_p$, a period system $P$, and a bounded $\mathbb C_p$-valued abstract measure $\mu$ on $\mathbb Z_p^\times$ (with scalar field $\mathbb C_p$), with the following properties. The period system consists of two nonzero complex numbers $\Omega_s$, indexed by $s\in\{+,-\}$, and a function $V:\{+,-\}\times\mathbb N\times\mathbb Q\to\overline{\mathbb Q}$ such that, for every $s$, every $0\le j\le k-2$, and every $r\in\mathbb Q$, $\iota(V(s,j,r))=[I_j(r)+\sigma_s(-1)^j I_j(-r)]/(2\Omega_s)$, where $\sigma_+=1$, $\sigma_-=-1$, and $I_j(r)=2\pi\int_{t>0}F(r+it)(r+it)^j\,dt$; the $\mathbb Z$-submodule of $\overline{\mathbb Q}$ spanned by all these $V(s,j,r)$ with $j\le k-2$ is finitely generated. Values of $V$ at larger $j$ are included in its domain but are subject to neither of these conditions. The element $\alpha$ satisfies $\lVert\alpha\rVert=1$ and $\alpha^2-\iota_p(a_p)\alpha+\iota_p(\varepsilon(p))p^{k-1}=0$. For every natural number $n\ge0$, every primitive $\overline{\mathbb Q}$-valued Dirichlet character $\chi$ modulo $m=p^n$, and every natural number $0\le j\le k-2$, there exist a continuous function $g:\mathbb Z_p^\times\to\mathbb C_p$ and $v\in\overline{\mathbb Q}$ such that, for every $x\in\mathbb Z_p^\times$, $g(x)=\iota_p(\chi(x\bmod p^n))\,\widetilde{x}^{\,j}$, where $\widetilde{x}$ is the image of the underlying $p$-adic integer in $\mathbb Q_p$ and then $\mathbb C_p$, and such that $\iota(v)=m^{j+1}j!\,L_{\chi,j}/[(-2\pi i)^jG_m(\chi^{-1})\Omega_{s(\chi,j)}]$ and $\mu(g)=\alpha^{-n}(1-\iota_p(\chi^{-1}(p))\iota_p(\varepsilon(p))p^{k-2-j}/\alpha)(1-\iota_p(\chi(p))p^j/\alpha)\iota_p(v)$. Here $\chi(p)$ and $\chi^{-1}(p)$ are evaluations at the residue class of $p$ in $\mathbb Z/m\mathbb Z$, $\chi^{-1}$ is the inverse Dirichlet character, $s(\chi,j)=+$ precisely when $\chi(-1)(-1)^j=1$ and is $-$ otherwise, $G_m(\psi)=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\psi(a))\exp(2\pi i\,a_0/m)$ with $a_0$ the canonical natural representative, $T_\chi(z)=G_m(\chi)^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))F(z+a_0/m)$, and $L_{\chi,j}=(2\pi)^{j+1}(j!)^{-1}\int_{t>0}T_\chi(it)t^j\,dt$. All the displayed complex integrals are the totalized Bochner integrals over $(0,\infty)$ with respect to Lebesgue measure; no separate integrability assumption is imposed in these definitions, and a nonintegrable integrand gives integral zero. Division and inversion are the field's total operations, so a zero denominator gives zero; $\alpha$ and the two periods are explicitly nonzero, whereas nonvanishing of the Gauss sums is not a separate hypothesis or conjunct. The interpolation quantifiers include $n=0$, hence modulus $1$, using exactly the same residue evaluations and sums, and include $j=0$; when $k=2$, $j=0$ is the only allowed exponent. The bounds ensure that the natural-number subtractions $k-1$, $k-2$, and $k-2-j$ in these assertions do not truncate, and $p$ prime ensures $m>0$. No coprimality condition between $p$ and $N$ is imposed, the eigenvalue relation includes primes dividing $N$, the assertion is existence rather than uniqueness, and no separate condition prescribing polynomial moments on residue disks is part of its conclusion.