Uniform boundedness of ordinary disk masses

For every prime natural number $p$, every positive natural number $N$, every natural number $k\ge 2$, and 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 the algebraic closure of $\mathbb Q$, the following holds. Suppose $f$ is a weight-$k$ cusp form for $\Gamma_1(N)$ (viewed as a subgroup of $\mathrm{GL}_2(\mathbb R)$), supplied with a Dirichlet character $\epsilon:\mathbb Z/N\mathbb Z\to\overline{\mathbb Q}$ and coefficients $c_n\in\overline{\mathbb Q}$ for every natural number $n$, such that its width-one $q$-expansion has coefficient $\iota(c_n)$ for every $n$, $c_1=1$, and, for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and every upper-half-plane point $z$, $f(\gamma z)=\iota(\epsilon(d))(cz+d)^k f(z)$. Suppose also that, for every prime natural number $\ell$ and every such $z$, $\ell^{-1}\sum_{b=0}^{\ell-1}f((z+b)/\ell)+\iota(\epsilon(\ell))\ell^{k-1}f(\ell z)=\iota(c_\ell)f(z)$, with integers passed to their residue classes when evaluating the character. Let a period system consist of nonzero complex numbers $\omega_s$ for both Boolean values $s$, and algebraic numbers $v_{s,j}(r)$ for every Boolean $s$, every natural number $j$, and every rational $r$. Define $\sigma_{\mathrm{true}}=1$, $\sigma_{\mathrm{false}}=-1$, and $I_j(r)=2\pi\int_{0}^{\infty}f(r+it)(r+it)^j\,dt$, with the integral interpreted as the complex Bochner integral over the positive real numbers (whose total definition returns zero if it is not integrable). Require $\iota(v_{s,j}(r))=(I_j(r)+\sigma_s(-1)^jI_j(-r))/(2\omega_s)$ for every $s,r$ and $0\le j\le k-2$, and require the $\mathbb Z$-submodule of $\overline{\mathbb Q}$ generated by all these $v_{s,j}(r)$ with $j\le k-2$ to be finitely generated; no corresponding comparison or finite-generation requirement is imposed on values with $j>k-2$. For every $\alpha\in\mathbb C_p$ satisfying $\lVert\alpha\rVert=1$ and $\alpha^2-\iota_p(c_p)\alpha+\iota_p(\epsilon(p))p^{k-1}=0$, there exists a real number $C\ge0$ such that, for both Boolean values $s$, every positive natural number $n$, and every integer $a$,

Here rational arguments are formed in $\mathbb Q$, powers of $p$ outside those arguments are interpreted in $\mathbb C_p$, and the norm is that of $\mathbb C_p$. The displayed expression is the stated disk moment at polynomial degree zero: its defining algebraic-symbol sum reduces to the single term $v_{s,0}(-a/m)$. The bound is simultaneous in $s,n,a$ and may depend on all the initially supplied data. It includes $a=0$, negative $a$, and integers divisible by $p$, without any coprimality or choice-of-residue-representative restriction. It includes $N=1$, $k=2$, and $p$ dividing $N$; for $k=2$ the factor $p^{k-2}$ is $1$, and for $n=1$ the second rational argument is $-a$. Depth $n=0$ is excluded, so the natural-number subtraction $n-1$ does not truncate below zero; likewise $k-2$ and $k-1$ do not truncate below zero. The norm-one condition implies $\alpha\ne0$, and primality implies $p\ne0$, so the displayed denominators are nonzero. The assertion is conditional on the existence of the specified eigenform, period system, and root and does not itself assert their existence.

Read-back model: GPT-6 (Codex).