15 problems

Bombieri–Dwork conjecture. The minimal differential equation of a $G$-function should come from geometry.

André's conjecture. $G$-functions are exactly the set of solutions over $\bar{\mathbb{Q}}[z]$ of a geometric differential equation over $\bar{\mathbb{Q}}$.

The paper considers D-finite power series and the classes of broad and strict G-functions. Siegel's conjecture. The classes of broad G-functions and strict G-functions should coinc…

Let $f(x)$ be a $G$-function, and suppose that its reduction modulo $p$ is defined for infinitely many primes $p$. Adamczewski–Delaygue conjecture. For almost all such primes $p$,…

Let ${\bf E}$, ${\bf G}$, and ${\bf D}$ denote the sets of values under consideration in the paper, with $\overline{\mathbb Q}$ the field of algebraic numbers. Intersection conject…

Let $\overline{\mathbb Q}(z)[\frac{d}{dz}]$ be the ring of linear differential operators over $\overline{\mathbb Q}(z)$. A differential operator is globally nilpotent if its $p$-cu…

Let $Q(theta)$ be a parity-symmetric $Q$-function, $Q(-theta)=Q(theta)$, and let $M^{-}$ be the $N\times N$ matrix with entries … Here $nu=2/(1+a)$ and $tilde\nu=2/(1-a)$. Boundary…

Let $Q(u)$ be the $Q$-function of the ground state, and let $Q_1(u)$ and $Q_2(u)$ be the $Q$-functions of arbitrary excited states $Psi_1$ and $Psi_2$. Suppose the norm formulas in…

Let $f(\phi)=\sum_{n=0}^{\infty}a_n\phi^n$ be a $G$-function with $a_n\in\mathbb{Q}$. Let $\phi_0$ be a singularity of its Picard–Fuchs equation $\mathscr{D}_f$, with maximally uni…

Let $\mathbb{K}$ be a number field, and let $L\in\mathbb{K}[z,\frac{{\rm d}}{{\rm d}z}]$ be a G-operator, meaning that its differential equation has a $G$-function solution for whi…

Stanley's conjecture. For every positive integer $r\ne 1$, the function $\mathfrak f_r$ is transcendental over $\mathbb C(z)$. The conjecture concerns a concrete family of $G$-func…

Let $F$ be a $G$-function and let $a/b$ be a rational number at which $F$ takes an irrational value. Rational approximation to $F(a/b)$ is measured by approximants with denominator…

Let $\mathbf{V}$ be the $\mathbf{S}$-module generated by the values of $E$-functions, and let ${\mathcal P}_e[1/\pi]$ denote the corresponding ring of exponential periods with…

Let $K$ be the coefficient field, let ${\mathcal P}_f$ be its finite places, and let ${\mathcal C}$ be the cyclotomic places. Suppose that … is a solution of a $q$-difference equat…

A $G$-module is a differential module over $K(x)/K$ whose uniform-part solutions are $G$-functions; equivalently, in a basis with associated system $Y'(x)=G(x)Y(x)$ and iterated ma…