139 problems

Let $n\geq1$ and let $x_1,\dots,x_n\in\mathbb{C}$ be $\mathbb{Q}$-linearly independent, meaning that no nonzero vector $(\lambda_1,\dots,\lambda_n)\in\mathbb{Q}^n$ satisfies…

Let $\mathbb{C}$ be equipped with its standard exponential map. Boris Zilber's axiomatization defines a unique exponential field satisfying the stated axioms and the Schanuel conje…

Let $X$ be a definable set in the structure $\mathbb R_{\exp}$, and consider the Pila–Wilkie counting function for arithmetic points of bounded height after removing the algebraic…

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…

For every positive integer $k$, consider the normalized odd zeta value $\zeta(2k+1)/\pi^{2k+1}$. Transcendence conjecture. All numbers … are transcendental. The source presents thi…

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…

An automatic number is a real number generated by an automatic sequence; an $S$-number is a transcendental number with finite, nonzero Diophantine exponent in the relevant classifi…

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

Let $M=[u:d4da0 \rightarrow G^n]$, with $u(1)=(R_1,\dots,R_n)$, be the 1-motive defined by the complex numbers $p_i$, $q_j$, and $t_{ij}$. Choose $p'_1,\dots,p'_{n'}$ and…

Let $a_1=(a_{11},a_{12})$ and $a_2=(a_{21},a_{22})$ be elements of $(\mathbb{Q}^{\times})^2$, and let $\mathcal{L}=\langle a_1,a_2\rangle$. Define … Assume that the rows and column…

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 $X$ be a smooth projective irreducible variety over $\overline{\mathbb{Q}}$, let $\beta$ be the de Rham--Betti comparison matrix in $\mathrm{GL}_N(\mathbb{C})$ under suitable b…

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…

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 $M$ be an $m \times n$ matrix with coefficients in $\mathscr{L}$ (or $\mathscr{L}_p$). Write … where $M_i \in M_{m \times n}(\mathbf{Q})$ and the $\lambda_i$ are elements of…

Let $\epsilon>0$, let $a_1,\ldots,a_n$ be positive integers, and let $b_1,\ldots,b_n$ be non-zero integers such that $a_1^{b_1}\cdots a_n^{b_n}\ne 1$. Lang–Waldschmidt conjecture.…

Let $n\ge2$ and let $\xi$ be a real number that is not algebraic of degree at most $n$. The height $H(\alpha)$ of an algebraic number is the height of its minimal defining polynomi…

Let $\mathcal{M}$ be the fixed category of motives, with de Rham and Betti fibre functors, motivic period algebra $\mathcal{P}^{\mathfrak{m}}$, and complex period map … A complex p…

Let $A$ be an abelian variety over $K=\mathbb C(S)$, where $S/\mathbb C$ is a curve, and let $F$ be the field of meromorphic functions on a disc in $S$. Let…

Let $x_1,x_2,y_1,y_2$ be complex numbers such that $x_1,x_2$ are linearly independent over the rational numbers and $y_1,y_2$ are linearly independent over the rational numbers. Fo…

Let $X$ be a smooth variety over $\overline{\mathbb{Q}}$. Let $P(X)$ be the period matrix of $X$, and let $\Gamma_X^{\operatorname{mot}}$ be the motivic Galois group of $X$ over…

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ of dimension $d$, and let $P$ be its period matrix, representing an isomorphism between…

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…

Three-logarithm conjecture. One has

For positive integers $s\ge 2$, define the Riemann zeta function by … The values at positive even integers are known to be rational multiples of powers of $\pi$, so the conjecture…