64 problems

An abelian surface $A_{/\mathbf{Q}}$ has conductor $N$, and write $\operatorname{End}_{\mathbf{Q}} A$ for its endomorphism ring over $\mathbf{Q}$. A QM abelian surface is an abelia…

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

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…

Brumer–Kramer paramodular conjecture. There is a bijection between $\mathcal A_N\cup\mathcal B_N$ and $\mathcal P_N$ such that

Brumer–Kramer conjecture. There exists such an $f$ satisfying

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…

An abelian surface over $\mathbf{Q}$ is expected to correspond to a Siegel modular form. Yoshida's conjecture. Abelian surfaces over $\mathbf{Q}$ should correspond to Siegel modula…

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…

Let $F$ be an irreducible holomorphic symplectic fourfold, and let $\lambda$ be a primitive divisor class of Beauville square zero on the boundary of the closure of the ample cone.…

Polynomiality conjecture. The quantity $c_1.c_2$ should be a polynomial in $p$ of a certain type, with its value known for small $p$.

Götsche's conjecture. Certain quasi-modular forms are the generating functions for the number of curves in the fixed linear system $|C|$.

Let $Th_-$ be the map from the moduli space of triples $(A,L,\vartheta)\in\mathcal{A}_2(3)^-$ to $\mathbb{P}^3_-$ introduced above. Its four components are functions on the releva…

Coleman's finiteness conjecture. The set of premodular pairs $(D,m)$ over $\mathbb{Q}$ is finite.

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 $A^{(k)}$ denote its $k$th symmetric product. Let $z\subset A^{(k)}$ be a cycle in the kernel of the summation map…

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…

Let $A$ be an abelian surface over a number field $K$. Write $A_0(A_K)$ for the group of zero-cycles of degree zero on $A_K$, and let $L(H^3(A),s)$ denote the $L$-function of the t…

Let $\boldsymbol{\tau}$ be a point on one of the modular or Shimura curves considered in the source, and let $\mathcal{L}^{lin}_{\boldsymbol{\tau}}$ be its lattice of singular line…

Let $\boldsymbol{\tau}$ be a period matrix on one of the Shimura curves under consideration, and let $\mathcal{L}_{\boldsymbol{\tau}}$ and $\mathcal{L}^{lin}_{\boldsymbol{\tau}}$ d…

Let $A/\mathbb{Q}$ be an abelian surface, and let $\pi_{A,ss}(x)$ denote the number of primes $p\leq x$ at which the reduction $\overline{A}_p$ is supersingular. Assume that $A$ be…

Let $f=\sum_{n\geq 1}a_nq^n$ be a non-CM newform of weight $2$, level $N$, and Nebentypus character $\epsilon$, and let $A_f/\mathbb{Q}$ be the absolutely simple abelian variety at…

Let $A/\mathbb{Q}$ be an abelian surface isogenous to a product of elliptic curves that are not $\overline{\mathbb{Q}}$-isogenous and are non-CM. For a subinterval $I\subset[-1,1]$…

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

Let $A$ be an Abelian surface over a $p$-adic local field $K$, let ${\mathcal A}$ be a semi-stable model over the ring of integers ${\mathcal O}_K$, and let ${\mathcal A}_v$ be its…