63 problems

Let $\mathcal E$ be an elliptic curve over $\mathbb{Q}(T)$ and let $A_{\mathcal E}(p)$ denote the average of the Frobenius traces over its fibers, as defined above. Nagao's conject…

Let $E(n)$ be the simply connected elliptic surface, and let $E(1)_{2,3}$ denote the result of applying $2$- and $3$-log transforms to $E(1)$; this is an exotic…

Let $E(n)$ be the simply connected elliptic surface, and let $E(n)_{p,q}$ be the complex surface obtained from $E(n)$ by a pair of $p$- and $q$-log transforms, where $n\geq 1$,…

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…

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

Kloosterman's conjecture. When

Let $X$ be a regular elliptic surface with geometric genus $p_g\ge 1$ and $r$ multiple fibres of multiplicities $m_1,\dots,m_r$. Assume that $X$ is simply connected, so that…

Minimum-height conjecture. These bounds are the correct values:

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

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…

Beilinson's conjecture. Equality always holds:

Nagao–Tate relation. Nagao's conjecture for $f$ predicts that the first residue in the displayed calculation is the rank of $\mathcal E(K_C)$, while Tate's conjecture for…

Silverman's analytic form of Nagao's conjecture.

Let $S$ be an elliptic surface with a section. Write $\operatorname{Hilb}^tS$ for the Hilbert scheme of length-$t$ subschemes of $S$, and let $\overline{\mathfrak M}_t$ be the comp…

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…

Let the compactified Coulomb branch of the rank two theory be an elliptic fibered surface, with singular fibers whose monodromies are represented by Dehn twists. The total mapping…

Toroidal compactification conjecture. There is a morphism

Let $V(L)$ be the genus $0$ Severi variety of a line bundle $L$ on a smooth projective algebraic surface, and consider the Severi curves arising from rational bisections of ellipti…

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…

Let $\E$ be the elliptic surface over $\mathbf{F}_q$ considered in the source, let $\overline{\mathbf{F}}_q$ be an algebraic closure, and let $\Fr_q$ denote Frobenius. Frobenius se…

Silverman's density conjecture. For all $b\in\mathbb{P}^1(\mathbb{Q})$ outside a set of density $0$ in $\mathbb{P}^1(\mathbb{Q})$,

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…