MTT: cusp forms, modular integrals, periods, and complex critical values

Qbar. This abbreviation denotes the chosen algebraic closure of the rational field, written $\overline{\mathbb Q}$; it has no parameters and does not itself choose an embedding into $\mathbb C$.

GammaOne. For every natural number $N$, this is the subgroup of $\mathrm{GL}_2(\mathbb R)$ obtained by mapping the integer determinant-one matrices $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ satisfying $a=d=1$ and $c=0$ in $\mathbb Z/N\mathbb Z$ into real invertible matrices. There is no positivity assumption on $N$: for $N=0$ these residue conditions are the integer equalities $a=d=1,c=0$, and for $N=1$ all determinant-one integer matrices satisfy them.

heckePrime. For natural numbers $k,l$, an arbitrary complex number $e$, an arbitrary function $f:\mathbb H\to\mathbb C$, and $z\in\mathbb H$, this defines $l^{-1}\sum_{b=0}^{l-1} f(U((z+b)/l))+e\,l^{\max(k-1,0)}f(U(lz))$, with all scalar arithmetic in $\mathbb C$. Here $\mathbb H$ is the upper half-plane and $U(w)$ is $w$ viewed as an upper-half-plane point when $\operatorname{Im}(w)>0$, and otherwise a fixed arbitrarily chosen upper-half-plane point. The definition assumes neither primality nor positivity of $l$, and no regularity of $f$. For $l=0$ the sum is empty and complex inversion of zero is zero; the remaining term is $e f(U(0))$ if $k=0$ or $1$, and zero if $k\ge2$, since $0^0=1$.

Eigenform. For arbitrary natural numbers $N,k$ and a unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$, this structure consists of a weight-$k$ holomorphic cusp form $f$ for the real image of the determinant-one integer matrices with $a=d=1,c=0$ modulo $N$, a Dirichlet character $\epsilon:\mathbb Z/N\mathbb Z\to\overline{\mathbb Q}$ (extended by zero on nonunits), and coefficients $A_n\in\overline{\mathbb Q}$ for every natural number $n$. Its required identities are that the $n$th coefficient of the width-one $q$-expansion of $f$ is $\iota(A_n)$ for every $n\ge0$, that $A_1=1$, that $f(\gamma z)=\iota(\epsilon(d))(cz+d)^k f(z)$ for every determinant-one integer matrix $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $c=0$ modulo $N$ and every $z\in\mathbb H$, and that $l^{-1}\sum_{b=0}^{l-1}f((z+b)/l)+\iota(\epsilon(l))l^{\max(k-1,0)}f(lz)=\iota(A_l)f(z)$ for every natural prime $l$ and every $z\in\mathbb H$. A cusp form here is invariant under the weight-$k$ slash action, holomorphic on $\mathbb H$, and vanishes at every cusp of that group; the width-one expansion uses the Taylor coefficients at zero of its cusp function in the parameter $q=e^{2\pi iz}$. No lower bounds on $N$ or $k$ are imposed, including $N=0,1$ and $k=0,1$; the declaration specifies data and conditions without asserting existence for every choice of parameters.

modularIntegral. For every function $f:\mathbb H\to\mathbb C$, complex polynomial $P$, and rational number $r$, this defines the complex number $2\pi\int_{(0,\infty)} f(r+it)P(r+it)\,dt$, where the integral is the Bochner integral with respect to real Lebesgue measure restricted to the positive half-line. The point $r+it$ is represented by the upper-half-plane conversion, which agrees with that point for all $t>0$ in the integration domain. No measurability, regularity, or integrability hypothesis is imposed; if the integrand is not integrable, the totalized Bochner integral is zero.

modularSymbol. For every function $f:\mathbb H\to\mathbb C$, natural number $j$, and rationals $a,m$, this defines $2\pi\int_{(0,\infty)} f(r+it)(m(r+it)+a)^j\,dt$ with $r=-a/m$ computed in $\mathbb Q$, and real Lebesgue measure. There are no integrability assumptions, and the integral is zero when the integrand is not integrable. The parameters include $j=0$, when the polynomial is identically one, and $m=0$, when rational division makes $r=0$ and the polynomial is the constant $a^j$ (including $0^0=1$).

sign. For a Boolean parameter $s$, this defines the integer $1$ when $s$ is true and the integer $-1$ when $s$ is false.

signedIntegral. For every function $f:\mathbb H\to\mathbb C$, Boolean $s$, natural number $j$, and rational $r$, this defines $\tfrac12\left(2\pi\int_{(0,\infty)}f(r+it)(r+it)^j\,dt+\sigma_s(-1)^j2\pi\int_{(0,\infty)}f(-r+it)(-r+it)^j\,dt\right)$, where $\sigma_s=1$ for true and $-1$ for false. Both integrals use real Lebesgue measure and each is independently zero if its integrand is not integrable; no analytic hypothesis on $f$ is required. The definition includes $j=0$ and $r=0$; at $r=0$ it equals the single integral multiplied by $(1+\sigma_s(-1)^j)/2$.

Periods. For a natural number $k$, a unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$, and an arbitrary function $f:\mathbb H\to\mathbb C$, this structure supplies complex numbers $\Omega_s\ne0$ for each Boolean $s$ and algebraic numbers $V_{s,j,r}$ for every Boolean $s$, natural number $j$, and rational $r$. For every such triple with $j\le\max(k-2,0)$ it requires $\iota(V_{s,j,r})=\left(2\pi\int_{(0,\infty)}f(r+it)(r+it)^j\,dt+\sigma_s(-1)^j2\pi\int_{(0,\infty)}f(-r+it)(-r+it)^j\,dt\right)/(2\Omega_s)$, where $\sigma_s$ is $1$ for true and $-1$ for false, and each Lebesgue Bochner integral is zero if its integrand is not integrable. It also requires that the $\mathbb Z$-submodule of $\overline{\mathbb Q}$ spanned by all $V_{s,j,r}$ satisfying that same bound on $j$ be finitely generated. No other comparison or finite-generation constraint is imposed on values outside that bound. In particular, for $k=0,1,2$ the bounded indices consist of $j=0$, rather than an empty range; no existence assertion or analytic assumption on $f$ is part of this structure declaration.

gaussSum. For a unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$, a natural number $m$ with a supplied assumption $m\ne0$, and a Dirichlet character $\chi$ modulo $m$ taking values in $\overline{\mathbb Q}$ and extended by zero on nonunits, this defines $\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))\exp(2\pi i\,\widetilde a/m)$, where $\widetilde a$ is the representative in $\{0,\ldots,m-1\}$. No primitivity or nonvanishing of this sum is required. Modulus one is included and its sole term gives the value $1$.

inverseTwist. For a unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$, an arbitrary function $f:\mathbb H\to\mathbb C$, a natural modulus $m$ with the assumption $m\ne0$, a Dirichlet character $\chi$ modulo $m$ valued in $\overline{\mathbb Q}$ and extended by zero on nonunits, and $z\in\mathbb H$, this defines $G^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))f(z+\widetilde a/m)$, where $G=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))\exp(2\pi i\,\widetilde a/m)$ and $\widetilde a\in\{0,\ldots,m-1\}$ is the standard representative. The translated arguments have positive imaginary part. There is no primitivity assumption and no assumption $G\ne0$; if $G=0$ the definition is zero by totalized complex inversion, while for $m=1$ it equals $f(z)$.

criticalLValue. For a unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$, an arbitrary function $f:\mathbb H\to\mathbb C$, a natural modulus $m$ with the assumption $m\ne0$, a Dirichlet character $\chi$ modulo $m$ valued in $\overline{\mathbb Q}$ and extended by zero on nonunits, and any natural number $j$, this defines $\frac{(2\pi)^{j+1}}{j!}\int_{(0,\infty)}\left[G^{-1}\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))f(it+\widetilde a/m)\right]t^j\,dt$, where $G=\sum_{a\in\mathbb Z/m\mathbb Z}\iota(\chi(a))\exp(2\pi i\,\widetilde a/m)$ and $\widetilde a$ is the representative in $\{0,\ldots,m-1\}$. The integral is the complex Bochner integral for real Lebesgue measure on the positive half-line and is zero if its integrand is not integrable. No cusp-form, weight, primitivity, convergence, or upper-bound-on-$j$ assumption is imposed. This includes $j=0$ and $m=1$; $j!$ is always nonzero, and if $G=0$ the twist and thus this value are zero by totalized inversion.