67 problems

Let $m(P)$ denote the Mahler measure of a Laurent polynomial $P$. Let $E$ be the elliptic curve … which has conductor $15$. Boyd and Rodriguez Villegas's conjecture. The Mahler mea…

Weak Lehmer conjecture. For each $s\in\mathbb N$, there exists $\varepsilon(s)>0$ such that for any irreducible monic polynomial $P$ with integer coefficients and $s(P)\leq s$, eit…

Let $\alpha$ be an algebraic integer of degree $d$, and consider the extremal algebraic integers for the house among the relevant nonreciprocal cases. Here nonreciprocal means that…

Margulis's conjecture. There is a function $\ell:\mathbb N\to\mathbb R^{>0}$ such that $m(p)\geq1+\ell\bigl(d(p)\bigr)$ for every non-cyclotomic monic polynomial…

Strong Lehmer conjecture. The optimal lower-bound constant is $c=\tau_{10}$.

Let $\beta$ be a parameter in the construction of generalized Turyn polynomials, and let the normalized Mahler measure refer to the Mahler measure divided by the square-root normal…

For odd integers $n$, let $\alpha_n$ be the Gaussian-period algebraic integer and let $C_{\kappa(n)}$ be the associated cyclovariety, with Mahler measures denoted by…

Let $\alpha_n$ be the Gaussian-period algebraic integer defined in the paper, and let $\mathrm{m}(\alpha_n)$ denote its Mahler measure. For a property of odd integers $n$, say that…

Limiting alternative-measure conjecture. The sequence has a limit, and

Let $c(P)$ be the alternative measure of a polynomial, and let $k\geq 2$. Consider the integer-exponent family $1+x+x^{n_2}+x^{n_3}+\cdots+x^{n_k}$ and take the iterated limit with…

Alternative-measure conjecture.

Let $P=(x+1)(y+1)(z+1)+t$, and let $f_7$ be the unique CM newform of weight $3$ and level $7$. Brunault's conjecture. The coefficients in the predicted Mahler-measure identity … ar…

Adelic hypersurface Lehmer conjecture. There exists a positive constant $c(n)$, depending only on $n$, such that

Let $P(x)\in\mathbb{Z}[x]$ be a monic irreducible polynomial, and let $m_{\mathbb{H}}(P)$ denote its quaternionic Mahler measure. Quaternionic Lehmer conjecture. There is a constan…

Let $\alpha$ be a nonzero algebraic integer that is not a root of unity, with all of its conjugates lying in the sector … where $0<\theta<90^\circ$. Let $c(\theta)>1$ be the functi…

Let $a,b\in\mathbb Q^\times$ be the coefficients in the conjectural identity … where $E_{45a2}$ is the elliptic curve indicated by its conductor label and $\chi_{-3}$ is the odd qu…

Let $\mathrm{m}(P)$ denote the logarithmic Mahler measure of a Laurent polynomial $P$ in two variables, and let $L(E_{15a8},s)$ denote the $L$-function associated with the elliptic…

Rogers' conjecture.

For a non-zero rational function $P\in\mathbb{C}(x_1,\dots,x_n)$, its logarithmic Mahler measure is … where $\mathbb{T}^n=\{(x_1,\dots,x_n)\in\mathbb{C}^n:|x_1|=\cdots=|x_n|=1\}$ i…

Let $p(x,y)\in\mathbb Z[x,y]$, and let $m,n$ be integers. In the relevant hyperbolic Dehn-filling applications, $p(x^m,x^n)$ is factored into irreducible factors. Lehmer's conjectu…

Let $P(x_1,\dots,x_n)\in\mathbb{C}[x_1,\dots,x_n]$ define a hypersurface in affine $n$-space, and complete affine space either to $\mathbb{P}^n$ or to $(\mathbb{P}^1)^n$. Let the t…

Let $\mathcal{L}$ be a $\mathbb{Z}^2$-periodic alternating link, with toroidally alternating quotient link $L$. Let $P(z,w)$ be the characteristic polynomial of the toroidal dimer…

Canonical-weight criterion conjecture. Up to $\operatorname{SL}(2,\mathbb{Z})$, the condition

Hauptmodul and monodromy-group conjecture. Then $k^n$ is a Hauptmodul for $\Gamma^b$,

Mahler-measure extension conjecture. The monotonic behaviour of the Mahler measure should extend to $|k|=0$.