25 problems

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…

Logarithmic-gap conjecture. One has

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…

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…

Frey–Mazur conjecture. If $E[p]$ and $E'[p]$ are isomorphic as Galois modules for some prime $p>17$, then $E$ and $E'$ are $\mathbb{Q}$-isogenous.

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…

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…

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 $K$ 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 $N$ 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…

Let $(X,\xi)$ be a quasi-polarized K3 surface over $k$. Let $(\Lambda_p,\lambda_p)$ and $(\Lambda^p,\lambda^p)$ be pointed lattices over $W(k)$ and $\hat{\mathbb{Z}}^p$, respective…

Let $E$ and $E'$ be elliptic curves over $\mathbb{Q}$, let $p$ be a prime, and let $E[p]$ and $E'[p]$ denote their $p$-torsion modules with Galois action by $G_{\mathbb{Q}}$. An is…

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$ over $\mathbb{C}$, and let $Z\subset\mathcal{A}_g$ be an irreducible algebraic s…

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$ over $\mathbb{C}$, let $Z\subset\mathcal{A}_g$ be an irreducible algebraic subva…

Generalized Lang–Trotter conjecture. The elliptic curves $E_1$ and $E_2$ satisfy

Let $\mathcal K$ be a local field of characteristic zero, and let $E/\mathcal K$ be an elliptic curve with a $p$-isogeny $\phi:E\to E'$ over $\mathcal K$. Put…

Let $F$ be a local field of characteristic zero, and let $E/F$ be an elliptic curve with a 2-isogeny $\phi:E\to E'$ over $F$. In the notation of the source, let $a,b\in F$ be the p…

Local $p$-parity formula. If $p\ne2$, then