40 problems

Wisde-prime infinitude conjecture. The set $\mathcal{W}$ is infinite. The paper gives computational evidence for many such primes and notes that this is equivalent to the infinitud…

Frey–Mazur conjecture. For all sufficiently large primes $p$, every $p$-congruence of elliptic curves comes from an isogeny.

Logarithmic-gap conjecture. One has

Let $(X,\operatorname{\sigma})$ be an interval exchange system (IES) with an invariant decomposition into ergodic probability measures, and let its rational invariants consist of t…

Let $\mathcal{O}_K$ be a discrete valuation domain with field of fractions $K$ and residue field $k$, and let $v$ be the normalized valuation on $K$. Let $E_1/K$ and $E_2/K$ be ell…

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…

Let $\Gamma\subset SL_2(\mathbb C)$ be a Zariski-dense subgroup. For a semisimple element $\gamma\in\Gamma$ of infinite order, let $\lambda$ be one of its eigenvalues; the associat…

Let $p$ be the prime parameter and let $A_{\max}$ denote the bound used by $\mathsf{dynamicPolz}(p)$, which increases in stages until the stopping condition…

Let $\mathbb{P}$ be the set of prime numbers, and let $\delta\colon\mathbb{P}\to\mathbb{Z}_{>0}$ assign to each prime $p$ the value $\delta(p)$. Prime-value surjectivity conjecture…

For each $n\in\mathbb{Z}_{>0}$, there should exist a prime $p$ such that, for every integer $m$ with $1\leq m\leq n$, there is a supersingular elliptic curve $E$ defined over…

Let $\mathcal{E}$ be the set of isomorphism classes of supersingular elliptic curves defined over $\overline{\mathbb{F}}_p$ for all primes $p$, and let…

Two elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$ are discriminant twins if they have the same minimal discriminant and the same conductor. They are semi-stable if they have no…

Let $(\mathcal{D}_{F_k},F_k)$, for $k\in\{0,1\}$, be Denjoy systems with invariants $(\rho_k,Q_k)$. For $A\subset\mathbb{T}$, write $\mathbb{Q}A\subset\mathbb{T}$ for the…

Weak Frey–Mazur conjecture. There is a constant $N_{E_0}$, depending only on $E_0$, such that if $p>N_{E_0}$ and

Let $\widetilde{N}_{5}^{\mathrm{tw}}(X)$ and $N_{5}^{\mathrm{tw}}(X)$ be the two counting functions for elliptic curves with a cyclic $5$-isogeny defined in the paper, and let…

Momose's conjecture. There are at most finitely many non-CM $k$-rational isogenies of Momose Type 3.

Let $E$ and $E'$ be rationally $2$-isogenous elliptic curves over $\mathbf{Q}$ such that … For a positive integer $m$, let $S_m$ be the set of anomalous primes of defect $(m+1,m)$.…

Let $E$ be an elliptic curve over a number field $F$. Say that $E$ is geometrically isogenous to an elliptic curve defined over $\mathbb{Q}$ if, after base change to an algebraic c…

Let $\mathcal{S}$ be a simple Shimura variety over $\mathbb{F}_q$, and let $C,D\subset \mathcal{S}$ be generically ordinary subvarieties having complementary dimension. Just-likely…

Let $K$ be a number field, let $\ell$ be a prime, and let $(A,\ell)$ be a strong counterexample to the local-global principle for isogenies of prime degree between abelian surfaces…

Let $K$ be a number field, let $A/K$ be an abelian surface, and let $\ell$ be a prime. Write $T_\ell A=\varprojlim_n A[\ell^n]$, let $G_\ell$ be the image of the mod-$\ell$ Galois…

Let $N$ be an integer and let $X$ be a K3 surface over $\overline{\mathbf{Q}}$. Consider the K3 surfaces isogenous to $X$ that can be defined over extensions of degree at most $N$.…

Let $\mathcal S$ be a simple Shimura variety over some base, and let $V,W\subseteq\mathcal S$ be generically ordinary subvarieties of complementary dimension. Unlikely intersection…

Let $X$ and $Y$ be abelian surfaces over $\bar{k}$ with finite height and $\operatorname{char}(k)\neq 2$. They are quasi-liftably isogenous if they are isogenous through suitable l…

Let $S$) and $S'$ be K3 surfaces over a finite field $k$ with $\operatorname{char}(k)=p>0$. A prime-to-$p$ derived isogeny is an equivalence of derived categories arising from a pr…