18 problems

Double-heptagon connection-point conjecture. The point $P$ is a connection point if and only if $N$ is even and $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Modulo-two orbit conjecture. An element $s=\frac{x}{y}\in\mathbb{Q}[\lambda_7]$ belongs to $H_7\cdot\infty$ if and only if $[\overline{x}:\overline{y}]$ belongs to the orbit of…

Brent's cusp-coefficient nonvanishing conjecture. There are corresponding polynomials $T_n\in\mathbb{Q}[x]$ satisfying $\tau_m(n)=T_n(m)$ for $m=3,4,\ldots$, together with the fact…

Let $H_{6,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{6,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $H_{4,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{4,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $a_n(x)$ be the irreducible factor occurring in the conjectured factorization of $A_n(x)$. Brent's root-location conjecture. If $n>1$, $a_n(\rho)=0$, and $\rho\neq\pm2$, then ……

Brent's nonvanishing conjecture. For each fixed integer $n\geq -1$ and every integer $m\geq 3$, one has

Brent's interpolation conjecture. For every integer $n>-2$, there exists a polynomial $A_n(x)\in\mathbb{Q}[x]$ such that $\alpha_n(m)=A_n(m)$ for $m=3,4,\ldots$, with…

Let $m\geq 3$, let $G(\lambda_m)$ be the Hecke group with $\lambda_m=2\cos(\pi/m)$, and write the Fourier expansion of the normalized invariant as … where…

Let $k\geq 5$, let $W$ be the relevant word set, let $\nu^{(k)}$ be the coefficient function, let $L_p\in\Lambda_p$ be the linear recurrence associated with the coefficients, and l…

Let $p=p_\lambda$ be the polynomial family considered in the paper, and let $\gamma(p)$ denote the quantity defined in the preceding observation. Parity formula for $p_\lambda$. Fo…

Let $G_k$ be the Hecke group with parameter $k$, let $W$ denote the words under consideration, and let $\nu_1,\ldots,\nu_8$ be the functions defined in Observation 1, using the Fib…

Prime Hecke noncongruence conjecture. The commutator subgroup of $H_q$ is not congruence.

Commutator congruence conjecture. The commutator subgroup of $H_q$ is congruence if and only if $H_q$ is commensurable with the modular group.

For the Hecke triangle group of type $(2,m,\infty)$, write … where $C_n,D_n\in\mathbb{Z}$ and $\gcd(C_n,D_n)=1$. Leo's first-occurrence conjecture. If a prime $p$ satisfying Leo's…

Consider the triangle group of type $(2,m,\infty)$ and write … where $C_n,D_n\in\mathbb{Z}$ and $\gcd(C_n,D_n)=1$. Leo's denominator-divisor conjecture. A prime $p$ divides some…

Let $S_{\mathfrak H(q)}(z)$ denote the generating function for the subgroup numbers of the Hecke group $\mathfrak H(q)$. For a positive integer $h$, define … Let $\alpha$ be a posi…

Let $q>9$, let $\lambda_q=2\cos(\pi/q)$, let $\mathbb Q(\lambda_q)$ be the field generated by $\lambda_q$, and let $G_q$ be the corresponding Hecke group acting on the real project…