74 problems

Algebraic-independence conjecture. The numbers $\ln p_1,\ldots,\ln p_m$ are algebraically independent. Equivalently, $\psi$ is an isomorphism.

Suppose $z_1,\ldots,z_n\in\mathbb{D}_p\subset\mathbb{C}_p$ are $\mathbb{Q}$-linearly independent, where $\mathbb{D}_p$ is the domain of the $p$-adic exponential function. Define…

Let $\alpha_1,\ldots,\alpha_n$ be nonzero algebraic numbers. Weak $p$-adic Schanuel conjecture. If $\log_p(\alpha_1),\ldots,\log_p(\alpha_n)$ are linearly independent over…

Let $z_1,\ldots,z_n\in\mathbb{H}^{+}$, write $\mathbf{z}=(z_1,\ldots,z_n)$, and let $j$ be the modular function. Let $\Sigma$ be the set of special points, and let…

Let $M=[u:\mathbb{Z}\to G^n]$ be the 1-motive with $u(1)=(R_1,\dots,R_n)$ defined by the points in the source. Let $k$ be the endomorphism field and let $\mathrm{LieBracket}$ and…

Let $k$ and $l$ be multiplicatively independent positive integers. Let $F\in K[[x]]$ be a power series over a number field $K$ that is both $k$-Mahler and $l$-Mahler. Loxton and Va…

Lang–Rohrlich conjecture. The value $\Gamma(z)$ is algebraic if and only if

Let $\Omega$ be a lattice in $\mathbb{C}$, let $k$ be the associated elliptic curve's endomorphism field, and let $g_2,g_3$ be its invariants. Let $p_1,\dots,p_n$ be $k$-linearly i…

Let $N$ be a level. Call $N$ standard if either $N=1,2,3$ or $N=p^n$ for a prime $p\ge 5$ and an integer $n\ge 1$; otherwise call it non-standard. Let $d(w,N)$ denote the dimension…

Exceptional-set power conjecture. If $\gamma\notin\mathbb{Q}$, then there exists $B\in\mathbb{N}$ such that

Transcendence conjecture for Euler's constant. The number $\gamma$ is transcendental. The source notes that even irrationality of $\gamma$ is not known, so this stronger conjecture…

Assume that $k$ is a CM field. Let $E_p(k)$ be the group of $p$-Weil numbers in $k$, let … where ${\mathbb R}(1)_v$ is the purely imaginary subspace at the complex place $v$, and l…

Schneider–Lang–Ramachandra's exponential conjecture. Under these assumptions, the determinant of $M$ is nonzero.

Let $H^+$ be the upper half-plane, and define … The groups $GL_2^+(\mathbb Q)$ and $B^+(\mathbb Q)$ act on $H^+$ by fractional linear transformations. Winkelmann's upper-half-plane…

Let $\alpha_1,\alpha_2\in\mathbb C$ be algebraic numbers with … Let … Two nonzero complex numbers are multiplicatively dependent if there are integers $(m,n)\ne(0,0)$ such that…

For each natural number $w$, let $Z_w$ be the $\mathbb Q$-vector space generated by multiple zeta values of weight $w$, let $Z_0=\mathbb Q$, and let…

Let $R_1^*$ denote the Flint Hills series and let $\tau=\frac{1-\frac{1}{2} u}{?}$ be... There exist rational constants $c_1,c_2 obreak\text{ in } obreak\bb Q$ such that … By Theor…

Let $T_p(z)$ denote the valuation generating function associated with a prime $p$. Let $p_1,p_2,\dots,p_m$ be distinct primes, and let $\alpha_1,\alpha_2,\dots,\alpha_m$ be non-zer…

Maximal transcendence-degree conjecture. The transcendence degree of the $\mathbb Q$-extension field generated by this set is maximal.

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent algebraic numbers, and let…

Let $\mathcal{A}$ be a commutative algebraic group over $\mathbb{Q}$ that is either a torus or an abelian variety, with Lie algebra $\mathfrak{a}$ over $\mathbb{Q}$ and exponential…

Let $\mathcal{E}$ be an elliptic curve with period lattice $\Omega$, invariants $g_2,g_3$, and endomorphism field $k$. Let $s,n\geq0$, let $\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}$…

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $p\in\mathbb{C}\smallsetminus\Omega$ with algebraic $\zeta(p)$, and let $\alpha$ be an algebraic…

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $s$ and $n$ be nonnegative integers. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent alg…

Split semi-elliptic conjecture. The transcendence degree of $K$ is at least $s+2n$, unless $2\pi\mathrm{i}\mathbb{Q}\subseteq\mathbb{Q}t_1+\cdots+\mathbb{Q}t_s$ and…