38 problems

Conjectured invariants of $S_q$. If $q$ is even, then

A number $t$ is represented as a ratio of two differences of fourth powers if there exist integers or rational numbers $a,b,c,d$ such that … with $c^4-d^4\neq 0$. Ratio of differen…

Let $a,b,c,d,e,fbinom{}{} inmathbb{Z}$ and consider the elliptic surface … where the coefficients $a_{i,j}$ are given by (ais). For $t\in\mathbb{Q}$, let $\mathcal{E}_t$ denote the…

Let $R$ be the ring of integers in an unramified extension $K$ of $\mathbb Q_p$, and let $\pi_R\colon X_R\to C_R$ be an elliptic surface over $R$ satisfying condition (Rat). Let…

Non-rationality conjecture. The specializations are not rational in $t$ with coefficients in $K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][v]$ and…

Let $\Phi:X\rightarrow\Sigma_g$ be a minimal elliptic surface, and let $H(X,\Phi)$ be the group of pairs of automorphisms preserving $\Phi$: … A semi-flat cscK current has fiber vo…

Existence and uniqueness conjecture. There is a semi-flat cscK current on $X$ whose regular fibers have volume $1$ and whose scalar curvature is $-3$; moreover, if $\omega_1$ and…

Let $S$ be an elliptic surface, let $C$ be a curve of genus $h$, and let $S_z$ denote the specified relative divisors in $S\times C$. For curve data $(\beta,n)$, insertions…

Toroidal compactification conjecture. There is a morphism

Period-preserving involution conjecture. The space $F_{1,1}$ admits a period-preserving birational involution $S\leftrightarrow S'$ such that

Let $F$ be an algebraically closed field, let $\E\to\mathbf{P}^1_F$ be a rational elliptic fibration with multiplicative fibers at $0$ and $1$ and an additive fiber at $\infty$, an…

A Mersenne prime is a prime number of the form … with $q$ prime. Mersenne-prime infinitude conjecture. There are infinitely many Mersenne primes. This conjecture is used to obtain…

Rank-zero fibre conjecture. There are infinitely many $b\in\mathbb{P}^1(\mathbb{Q})$ such that

Non-isotrivial elliptic-surface rank conjecture. If $\pi$ is non-isotrivial, then both $\mathscr{J}(X,\pi)$ and $\mathscr{N}(X,\pi)$ are infinite.

Equidistribution of root numbers conjecture. Both $S_{-1}$ and $S_1$ have natural density $1/2$ in $\mathbb{N}$. This is the equivalent sign-equidistribution formulation of the con…

Averages of root numbers conjecture. The average root number in this family is zero. Helfgott gives strong support and proves the assertion for several elliptic surfaces under suit…

Positivity of the rank. There is $c>0$ such that for all sufficiently large $x$, $N(x)\ge c\cdot x$. This conjecture is motivated by work of Helfgott and, together with a parity st…

Let $W$ be a rational Weierstrass fibration, let $X$ be its minimal resolution, and let $H$ be an arbitrary ample line bundle. Miranda's conjecture. The pair $(W,H)$ is asymptotica…

Let $f:M\to\Sigma$ be an elliptic bundle over a Riemann surface $\Sigma$ of genus at least $2$, and let $\omega(t)$ solve the Chern-Ricci flow on $M$ from an arbitrary Gauduchon me…

Let $X$ be a minimal surface of Kodaira dimension $1$ with an elliptic fibration … Suppose that $P_g(X)=0$ and $q(X)=1$. Elliptic-fibration decomposability conjecture. Then $D(X)$…

Rational-point conjecture. The only ${\mathbb Q}$-points on $\widetilde{Z}(17,1)$ lie above one of the listed curves in $(x,y)$, the curve $T=0$, or one of the points in the indica…

Average-rank conjecture. One has

For positive integers $m$ and $n$, let $\mathcal{S}_{\ell;m,n}$ be the finite set of elliptic surfaces over $\mathbb{F}_\ell$ of the form $y^2=x^3+A(T)x+B(T)$ with $\deg(A)=m$ and…

For positive integers $d$ and $M$, let … where $\mu(p)$ is the Mahler measure, and define … Here $\mathcal{S}_{m,n}=\mathcal{S}_{m,n}(M)$ is the family of elliptic surfaces with fi…

Let $\pi_2$ be the elliptic fibration constructed from a point of ${\mathcal M}'_{K,1}$, let $\sigma_0$ be its zero section, and let $\phi$ be the map associated with the correspon…