36 problems

Let $K$ be an imaginary quadratic field. A Bianchi newform over $K$ is a weight-$2$ cuspidal eigenform of level $[?]$ with rational Hecke eigenvalues; here $[?]$ denotes the ideal…

Joyce's conjecture. For every $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $alphain N(X)$, there is an invariant $J^{\alpha}(\sigma)inLambda$ such that $J^{\alpha}(\sigma)$ does not de…

Let $X$ and $\widetilde{X}$ be nonsingular projective toric $4$-folds, and let $\psi:\widetilde{X}\rightarrow X$ be a 2-blow-up, meaning an equivariant blow-up along a $T_N$-invari…

Let $A$ be an abelian surface, and let $F$ be a simple vector bundle on $A$ with Mukai vector $v:=v(F)$. Suppose that $c_1(F)$ is symmetric. For a positive integer $d$, define … wh…

Let $S$ be a symplectic surface, and let $\beta\in H_2(S,\mathbb{Z})$ be an effective curve class. For every positive integer $k$ dividing $\beta$, let … be a degree-preserving…

Hasse–Weil conjecture. The function $\zeta_X(s)$ extends to a meromorphic function of $\mathbf C$. There exists a positive real number $A\in\mathbf R^{>0}$, non-zero rational funct…

Square-root error conjecture. For fixed $t$, $m$, and $I\subset[-1,1]$,

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 $C$ be a curve class, let $g_0$ be fixed, and define the FLS generating series … Let $R^{FLS}_{g_0,C}(q)$ denote the corresponding refined tropical FLS generating series, and s…

Let $C$ be a curve class, let $g_0$ be fixed, and define the generating series … Let $R_{g_0,C}(q)$ denote the corresponding refined tropical generating series, and set $q=e^{iu}$.…

Piecewise linearity conjecture. The Seshadri function of $A$ is piecewise linear precisely in these cases.

Exceptional-set conjecture.

Let $p\ge 5$ and let $\ell\ne p$ be prime. Let $\mathcal{G}^{SS}_{2}(\ell,p)$ be the superspecial isogeny graph, with normalized adjacency eigenvalues … where $m=|SS_2(p)|$, and se…

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 $A$ be an abelian surface, let $\beta\in H_2(A,\mathbb{Z})$ be effective, and for each divisor $k\mid\beta$ choose an abelian surface $A_k$ and a parallel-transport morphism…

Let $A$ be an Abelian surface of Picard rank one, let $\tau$ be the involution defining the quotient, and let $C\subset A$ be an integral $\tau$-invariant curve of geometric genus…

Uniqueness conjecture. For every ample $\mathbb Q$-line bundle $L$ on $X$, there exists at most one irreducible curve $C$ that is submaximal for $L$ if and only if either

Let $\mathcal{N}_{2,\mathbb{Q}}(\mathcal{B})$ count principally polarized abelian surfaces $A=\operatorname{Jac}(X)$, where $X$ is a stable genus $2$ curve over $\mathbb{Z}$ satisf…

Local root-number formula. The local root number of the Jacobian satisfies

Let $A$ be an abelian surface over a number field $k$, and let $\operatorname{ST}_{A}\subset\operatorname{USp}(4)$ be its Sato–Tate group. For each prime $\mathfrak{p}$ of good red…

Let $B$ be a polarized Abelian surface, let $C$ be a hyperelliptic curve, and let $f:C\to B$ be a degree-$1$ morphism with image $\bar{C}$ such that the polarization $[\bar{C}]$ is…

Let $A$ range over isogeny classes of abelian surfaces over $\mathbf{Q}$ of conductor $N$ with $\operatorname{End}_{\mathbf{Q}} A=\mathbf{Z}$. Let $K(N)$ be the paramodular group ……

Let $E_1$ and $E_2$ be nonsingular elliptic curves, let $\mathsf{A}=E_1\times E_2$ be fibered over $E_1$, and let ${\mathcal A}_{g,n}(\gamma_1,\ldots,\gamma_n)$ denote the reduced…

Let $\mathcal{A}/\mathbb{Q}$ be an abelian surface of conductor $N$ with $\operatorname{End}_{\mathbb{Q}}\mathcal{A}=\mathbb{Z}$. Let $F$ be a weight $2$ cuspidal paramodular newfo…

Gross–Popescu's conjecture. The homogeneous ideal of $A$ in this embedding is generated by quadrics and cubics.