15 problems

Let $E$ be an elliptic curve over $\mathbb{Q}(t)$, let $E(\mathbb{A}^1)$ denote the elliptic curve over the function field $\mathbb{Q}(t)$, and let $E(t)$ denote its specialization…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts. A $(0,1)$-substitution map is an operator $\varphi$ on…

Let $K$ be a number field, let $E/K(T)$ be a nonisotrivial elliptic curve with generic Mordell–Weil rank $r={\operatorname{rank}} E(K(T))$, and let $\mathcal B$ be the finite set o…

Non-jumping fiber conjecture. The set $\mathcal N(E)$ is infinite, i.e. there are infinitely many $t\in\mathbb{Q}$ for which $E_t(\mathbb{Q})$ has rank $r$.

Let $X$ be as in the cited theorem of Kollár--Tian, with generic fiber $X_K$ and integer $n$ as in that theorem. Write $C\!H_2(X)$ and $C\!H_1(X_K)$ for the relevant Chow groups, a…

Let $\mathcal{E}$ be an elliptic curve over the function field $\Q(T)$ with nonconstant $j$-invariant. For $t\in\Q$, let $\mathcal{E}_t$ denote its specialization when it is an ell…

Let $A\to T^0$ be a non-isotrivial family of abelian varieties over a number field $K$, with generic fiber $A_{\eta}$ simple of dimension at least two. A section $P:T^0\to A$ is no…

Beckmann–Black lifting conjecture. Every $H$-extension of a number field $K$ is a specialization of a $K$-regular $H$-extension of $K(t)$.

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\mathrm{M}_K$ be the abelian group freely generated by isomorphism classes of smooth proper…

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{GT}_L)$ and $K_0(\mathrm{sGT}_L)$ denote the Grothendieck grou…

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{sGT}_L)$ be the Grothendieck group generated by bounded derive…

Let $\mathcal{F}$ be a non-isotrivial family of elliptic curves over $\mathbb{Q}(t)$, with generic rank $r_{\mathcal{F}}$. For all but a zero-density subset of rational parameters…

Dwork's specialization conjecture. The polygon $\mathrm{NP}_{\log,0}(D)$ is above $\mathrm{NP}_{\log,t}(D_t)$.

Let $K$ be a number field, let $C$ be a curve over $K$, and put $F=K(C)$. Let $\varphi(x,y)$ be a Hénon map over $F$, and let $P,Q\in\mathbb{A}^2(F)$ have distinct orbits under…

Geometric generic-rank conjecture. Lemma should remain valid with $B({\mathcal O}_{\overline{K}})$ replaced by $B({\overline{K}})$.