MTT: p-adic measures, disk moments, and interpolation

UnitMeasure. For every natural prime $p$ (with a supplied proof of primality), this is the type of abstract $\mathbb C_p$-valued, $\mathbb C_p$-linear measures on the group $\mathbb Z_p^\times$ of units in the $p$-adic integers: bounded continuous linear functionals on continuous $\mathbb C_p$-valued functions on that group. No positivity or probability normalization is imposed.

coordinate. For a natural prime $p$ and a unit $x\in\mathbb Z_p^\times$, this is its underlying $p$-adic integer, included first in $\mathbb Q_p$ and then by the specified algebra map into $\mathbb C_p$; denote it by $c(x)$.

IsOrdinaryRoot. Given a natural prime $p$, natural numbers $N,k$, a ring embedding $\iota:\overline{\mathbb Q}\to\mathbb C$, an Eigenform datum $f$ of level $N$ and weight $k$, a ring embedding $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, and $\alpha\in\mathbb C_p$, this predicate is the conjunction $\|\alpha\|=1$ and $\alpha^2-\iota_p(a_p)\alpha+\iota_p(\varepsilon(p))p^{\max(k-1,0)}=0$. Here the Eigenform datum consists of a cusp form $F$ for the real matrix image of $\Gamma_1(N)$ of integer weight $k$, a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $a_n\in\overline{\mathbb Q}$ whose images under $\iota$ are the coefficients of the period-one $q$-expansion of $F$, with $a_1=1$. It includes the law $F(\gamma z)=\iota(\varepsilon(d))(cz+d)^kF(z)$ for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and every upper-half-plane point $z$, and, for every prime $\ell$, the identity $\ell^{-1}\sum_{b=0}^{\ell-1}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. Neither $N$ nor $k$ has a positive lower bound in this declaration; there is no condition that $p$ be coprime to $N$.

algebraicSymbol. For a natural number $k$, a ring embedding $\iota:\overline{\mathbb Q}\to\mathbb C$, an arbitrary function $F$ on the upper half-plane, a Periods datum $P$ for $(k,\iota,F)$, a Boolean $s$, natural $j$, and rational $a,m$, the result is $\sum_{t=0}^j\binom jt m^t a^{j-t}V(s,t,-a/m)\in\overline{\mathbb Q}$. The datum $P$ supplies nonzero complex numbers $\omega_s$ and values $V(s,t,r)\in\overline{\mathbb Q}$ for every Boolean $s$, natural $t$, and rational $r$. For $t\le\max(k-2,0)$ it requires $\iota(V(s,t,r))=(I_t(r)+\sigma_s(-1)^tI_t(-r))/(2\omega_s)$, where $\sigma_{\mathrm{true}}=1$, $\sigma_{\mathrm{false}}=-1$, and $I_t(r)=2\pi\int_0^\infty F(r+iu)(r+iu)^t\,du$. The integer submodule spanned by all these values with $t\le\max(k-2,0)$ must be finitely generated. No comparison is imposed on larger $t$. No bound on $j$ or nonzero condition on $m$ is required here: $m=0$ makes $-a/m=0$ by totalized field division, and zeroth powers, including $0^0$, are $1$. The integrals are totalized Bochner integrals, equal to zero when the integrand is not integrable.

diskMoment. Given a natural prime $p$, natural $N,k$, embeddings $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, an Eigenform datum $f=(F,\varepsilon,(a_r))$ of level $N$, weight $k$ and embedding $\iota$, a Periods datum $P$ for $F$, any $\alpha\in\mathbb C_p$, Boolean $s$, natural $j,n$, and integer $a$, this is $\alpha^{-n}\iota_p(S(s,j,a,p^n))-\iota_p(\varepsilon(p))p^{\max(k-2,0)}(\alpha^{n+1})^{-1}\iota_p(S(s,j,a,p^{\max(n-1,0)}))$, where $\alpha^{-n}$ here means $(\alpha^n)^{-1}$ and $S(s,j,a,m)=\sum_{t=0}^j\binom jt m^ta^{j-t}V(s,t,-a/m)$. The Eigenform datum requires the normalized algebraic period-one Fourier expansion of a $\Gamma_1(N)$ cusp form, the $\Gamma_0(N)$ transformation law with character $\varepsilon(d)$, and the prime eigenrelations $\ell^{-1}\sum_{b<\ell}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. The Periods datum supplies nonzero $\omega_s$ and algebraic $V(s,t,r)$ with $\iota(V(s,t,r))=(I_t(r)+\sigma_s(-1)^tI_t(-r))/(2\omega_s)$ for $t\le\max(k-2,0)$, where $I_t(r)=2\pi\int_0^\infty F(r+iu)(r+iu)^tdu$ and $\sigma_s$ is $1$ for true and $-1$ for false; those bounded-degree values span a finitely generated integer submodule. These are totalized Bochner integrals. This definition imposes no ordinary-root condition, no coprimality on $a$, no degree bound, and no positive-depth condition. In particular $n=0$ uses $p^0=1$ in both symbols, and $\alpha=0$ is allowed with $0^{-1}=0$ and $0^0=1$.

diskFunction. For a natural prime $p$, natural $n,j$, integer $a$, and unit $x\in\mathbb Z_p^\times$, this is $c(x)^j$ when the underlying $p$-adic integer of $x$ reduces to $a$ modulo $p^n$, and is zero otherwise; $c(x)$ is its image under $\mathbb Z_p\hookrightarrow\mathbb Q_p\hookrightarrow\mathbb C_p$. At $n=0$ the modulus is one, so the condition holds for every $x$. At $j=0$ the function is the indicator of that residue class. There is no coprimality assumption on $a$; at positive depth a residue class divisible by $p$ contains no such units.

RealizesMoments. For a natural prime $p$, natural $N,k$, embeddings $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, an Eigenform datum $f=(F,\varepsilon,(a_r))$, a Periods datum $(\omega,V)$ for $F$, any $\alpha\in\mathbb C_p$, Boolean $s$, and bounded $\mathbb C_p$-linear functional $\mu$ on continuous functions $\mathbb Z_p^\times\to\mathbb C_p$, this predicate says: for every positive natural $n$, every integer $a$ coprime to $p$, and every natural $j\le\max(k-2,0)$, there exists a continuous function $g$ equal at every unit $x$ to $c(x)^j$ if $x\equiv a\pmod{p^n}$ and zero otherwise, such that $\mu(g)=(\alpha^n)^{-1}\iota_p(S(s,j,a,p^n))-\iota_p(\varepsilon(p))p^{\max(k-2,0)}(\alpha^{n+1})^{-1}\iota_p(S(s,j,a,p^{n-1}))$. Here $c(x)$ is the image of $x$ in $\mathbb C_p$, and $S(s,j,a,m)=\sum_{t=0}^j\binom jt m^ta^{j-t}V(s,t,-a/m)$. The Eigenform datum requires a $\Gamma_1(N)$ cusp form of weight $k$, Fourier coefficients $\iota(a_r)$ with $a_1=1$, character law $F(\gamma z)=\iota(\varepsilon(d))(cz+d)^kF(z)$ for $\gamma\in\Gamma_0(N)$, and prime eigenrelations $\ell^{-1}\sum_{b<\ell}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. The Periods datum requires nonzero $\omega_s$ and $\iota(V(s,t,r))=(I_t(r)+\sigma_s(-1)^tI_t(-r))/(2\omega_s)$ for $t\le\max(k-2,0)$, with $I_t(r)=2\pi\int_0^\infty F(r+iu)(r+iu)^tdu$, $\sigma_{\mathrm{true}}=1$, $\sigma_{\mathrm{false}}=-1$, and a finitely generated integer span of those values; the integrals are totalized Bochner integrals. No ordinary-root or nonzero condition on $\alpha$ occurs. Weights zero and one still require degree zero moments. The predicate concerns the supplied $\mu$, rather than asserting that a measure exists.

specialFunction. Given a natural prime $p$, a ring embedding $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, natural $n,j$, a $\overline{\mathbb Q}$-valued Dirichlet character $\chi$ modulo $p^n$, and a unit $x\in\mathbb Z_p^\times$, this is $\iota_p(\chi(x\bmod p^n))c(x)^j$, with $c(x)$ the image of $x$ in $\mathbb C_p$. There is no primitive-character assumption or positive-depth assumption; $n=0$ uses the modulus-one character and $j=0$ uses the zeroth power $1$.

criticalSign. For any natural modulus $m$, any $\overline{\mathbb Q}$-valued Dirichlet character $\chi$ modulo $m$, and any natural $j$, this Boolean is true precisely when $\chi(-1)(-1)^j=1$ in $\overline{\mathbb Q}$, and is false otherwise. This declaration does not require positive modulus or primitivity; it includes $m=0$, $m=1$, and $j=0$.

eulerMultiplier. For a natural prime $p$, natural $N,k$, embeddings $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, an Eigenform datum $f=(F,\varepsilon,(a_r))$, any $\alpha\in\mathbb C_p$, natural $n,j$, and a Dirichlet character $\chi$ modulo $p^n$ valued in $\overline{\mathbb Q}$, this is $(\alpha^n)^{-1}(1-\iota_p(\chi^{-1}(p\bmod p^n))\iota_p(\varepsilon(p))p^{\max(k-2-j,0)}/\alpha)(1-\iota_p(\chi(p\bmod p^n))p^j/\alpha)$. The Eigenform datum consists of a weight-$k$ $\Gamma_1(N)$ cusp form with normalized algebraic Fourier coefficients, the $\Gamma_0(N)$ character law $F(\gamma z)=\iota(\varepsilon(d))(cz+d)^kF(z)$, and prime eigenrelations $\ell^{-1}\sum_{b<\ell}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. The character is evaluated at the displayed modulus, with no requirement of primitivity and no change of modulus. For $n>0$, its value and the inverse character's value at $p$ are zero, so the two parenthesized factors are one. For $n=0$ the modulus is one. Neither a bound on $j$ nor an ordinary-root or nonzero condition on $\alpha$ is imposed; inverse zero is zero, so $\alpha=0$ yields zero when $n>0$ and one when $n=0$.

normalizedCriticalValue. For a natural prime $p$, natural $N,k$, an embedding $\iota:\overline{\mathbb Q}\to\mathbb C$, an Eigenform datum $f=(F,\varepsilon,(a_r))$, an arbitrary function $\omega:\{\mathrm{false},\mathrm{true}\}\to\mathbb C$, natural $n,j$, and an algebraic-valued Dirichlet character $\chi$ modulo $m=p^n$, this is $m^{j+1}j!\,L/((-2\pi i)^jG(\chi^{-1})\omega_s)$, where $s$ is true exactly when $\chi(-1)(-1)^j=1$, $G(\psi)=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\psi(a))\exp(2\pi i\widetilde a/m)$ with $\widetilde a\in\{0,\ldots,m-1\}$, and $L=(2\pi)^{j+1}/j!\int_0^\infty[G(\chi)^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))F(it+\widetilde a/m)]t^jdt$. The Eigenform datum requires a weight-$k$ $\Gamma_1(N)$ cusp form, Fourier coefficients $\iota(a_r)$ with $a_1=1$, the $\Gamma_0(N)$ law with multiplier $\iota(\varepsilon(d))(cz+d)^k$, and prime eigenrelations $\ell^{-1}\sum_{b<\ell}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. There is no requirement that $\omega$ be a period system or nonzero, that $\chi$ be primitive, or that $j$ be in a critical range. The modulus-one case $n=0$ is included. All divisions use inverse zero equal to zero, and the integral is the totalized Bochner integral, equal to zero for a nonintegrable integrand.

Interpolates. For a natural prime $p$, natural $N,k$, embeddings $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, an Eigenform datum $f=(F,\varepsilon,(a_r))$, any function $\omega:\{\mathrm{false},\mathrm{true}\}\to\mathbb C$, any $\alpha\in\mathbb C_p$, and a bounded $\mathbb C_p$-linear functional $\mu$ on continuous functions on $\mathbb Z_p^\times$, this predicate says that for every natural $n$, every primitive algebraic-valued Dirichlet character $\chi$ modulo $m=p^n$, and every natural $j\le\max(k-2,0)$, there exist a continuous function $g:\mathbb Z_p^\times\to\mathbb C_p$ and $v\in\overline{\mathbb Q}$ such that $g(x)=\iota_p(\chi(x\bmod m))c(x)^j$ for every unit $x$, $\iota(v)=m^{j+1}j!L/((-2\pi i)^jG(\chi^{-1})\omega_s)$, and $\mu(g)=(\alpha^n)^{-1}(1-\iota_p(\chi^{-1}(p\bmod m))\iota_p(\varepsilon(p))p^{\max(k-2-j,0)}/\alpha)(1-\iota_p(\chi(p\bmod m))p^j/\alpha)\iota_p(v)$. Here $c(x)$ is the image of $x$ in $\mathbb C_p$, $s$ is true exactly when $\chi(-1)(-1)^j=1$, $G(\psi)=\sum_{a\bmod m}\iota(\psi(a))\exp(2\pi i\widetilde a/m)$, and $L=(2\pi)^{j+1}/j!\int_0^\infty[G(\chi)^{-1}\sum_{a\bmod m}\iota(\chi(a))F(it+\widetilde a/m)]t^jdt$, with least nonnegative representatives $\widetilde a$ and totalized Bochner integration. The Eigenform datum requires a weight-$k$ $\Gamma_1(N)$ cusp form with algebraic Fourier coefficients $\iota(a_r)$, $a_1=1$, the law $F(\gamma z)=\iota(\varepsilon(d))(cz+d)^kF(z)$ for $\gamma\in\Gamma_0(N)$, and prime eigenrelations $\ell^{-1}\sum_{b<\ell}F((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{\max(k-1,0)}F(\ell z)=\iota(a_\ell)F(z)$. The predicate imposes neither a period-system condition on $\omega$ nor an ordinary-root condition on $\alpha$; both may vanish, and divisions by zero are zero. It includes $n=0$ and the primitive modulus-one character, and includes degree zero even for $k=0$ or $1$. At positive depth the character values at $p$ vanish. The existential algebraic $v$ requires the displayed normalized complex expression to lie in the image of $\iota$. The statement concerns a supplied $\mu$, without asserting existence of such a measure, and imposes no condition for nonprimitive characters.