334 problems

Morton–Silverman conjecture. There exists a constant $C=C(d,N,[k:{\mathbb Q}])>0$ such that every degree $d$ endomorphism $f$ of ${\mathbb P}^N$ over $k$ satisfies

Let $X$ be a quasi-projective variety defined over $\mathbb{C}$, let $\Phi$ be an endomorphism of $X$, let $V\subset X$ be a closed subvariety, and let $x\in X(\mathbb{C})$. Dynami…

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ be a dominant rational self-map defined over $\mathbb{Q}$, and let $P=[x_0:y_0:z_0]$ be a generic rational point. Write $f^n(P)$ f…

Let $f(x)=x^2-1$, and for an integer $n\geqslant 2$ let $P(n)$ denote the set of prime numbers arising in the associated sequence defined in the paper. Density-zero conjecture. For…

Let $f$ be a quadratic polynomial defined over ${\mathbb Q}$, and let a rational periodic point have exact period $N$. Poonen's conjecture. Such a polynomial cannot have a rational…

Flynn–Poonen–Schaefer conjecture. There is no quadratic polynomial $\varphi_{2,c}(z)=z^2+c\in\mathbb Q[z]$ with a rational point of exact period $n$.

Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ map over $\overline{\mathbb{Q}}$, and let $c$ range over its critical points, counted with multiplicity. Define the critical hei…

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ de…

Let $k$ be an algebraically closed field, let $V$ be a variety over $k$, and let $(V,\phi)$ be a $\sigma$-variety over $k$, with $\phi$ the self-map defining its difference structu…

Hutz's conjecture. There is no even degree $d>2$ and no $c\in\mathbb Q$ such that $\varphi_{d,c}$ has a rational point of exact period $n$. Moreover,

Let $X$ be a projective variety over $k$, and let $\phi:X\rightarrow X$ be a polarized endomorphism. The forward orbit of $x\in X(k)$ is $\{\phi^m(x):m\geq 0\}$. Zhang's Zariski de…

Let $X$ be a variety and let $\Phi:X\to X$ be a dominant map. A point of $X$ is preperiodic for $\Phi$ if its forward orbit under $\Phi$ is finite, and a subvariety $Y$ is preperio…

Let $K$ be a number field, let $K^{\mathrm{ab}}$ be its maximal abelian extension, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a polynomial map of degree $d\geq2$ defined over $K$, an…

Let $f,g$ be polynomials of the same degree $d\geq 2$, and let $\operatorname{Prep}(f)$ denote the set of preperiodic points of $f$. DeMarco–Krieger–Ye's conjecture. There should e…

Let $K/\mathbb{Q}$ be a quadratic field, let $f\in K[z]$ be a quadratic polynomial, and let $\operatorname{PrePer}(f,K)$ denote its set of $K$-rational preperiodic points. Doyle's…

Let $f:X\to X$ be a morphism of a smooth projective variety over $\overline{\mathbb{Q}}$, let $D$ be an ample eigendivisor satisfying $f^*D\sim\delta D$ for some $\delta>1$, and le…

For a polynomial $f\in\mathbb{Q}[X]$ and a prime $p$, let $\bar{f}_p\in\mathbb{F}_p[X]$ be its reduction modulo $p$. A polynomial over a field is dynamically irreducible if every i…

Let $d\geq 2$, let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ dynamical system, let $h_{\mathcal{M}}$ be a height on the moduli space $\mathcal{M}_d$ of degree-$d$ dynamical s…

Let $f:X\to X$ be a polarized endomorphism of a smooth projective variety over a field of characteristic zero, and let $Z\subset X$ be a subvariety containing a Zariski dense set o…

Let $K$ be a complete discretely valued field. An extension $L/K$ is weakly totally ramified if it is separable and has the relevant ramification property while preserving the resi…

Adelic Zariski dense orbit conjecture. The stated adelic-open-set conclusion holds.

Let $S$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$, let $\Phi:S\times\mathbb{P}^N\to S\times\mathbb{P}^N$ be an algebraic family of endomorphisms of degree…

Let $f$ be the post-critically finite quadratic polynomial under consideration, and let $G_n(f)$ and $M_n(f)$ denote, respectively, its level-$n$ Galois group and even Markov group…

A rational map is eventually stable when its iterated arithmetic behavior satisfies the eventual stability property described in the paper. The conjecture concerns rational maps de…

Let $X/\overline{\mathbb{Q}}$ be a smooth projective variety, let $f:X\to X$ be a dominant morphism with $\delta(f)>1$, and define … with the infimum of the empty set defined as…