22 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…

Let the pure theta block $\Theta_f$ be a holomorphic Jacobi form of weight $k$ and index $m$ with vanishing order $1$ in $q$, where $k,m\in\mathbb Z_{>0}$. Define the nearly holomo…

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 $\mathcal{A}/\mathbb{Q}$ 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…

The characterization of Eisenstein-type forms. The subspace of $M_{a,b}(\operatorname{SO}(\Lambda),\vartheta_d)$ spanned by forms of Eisenstein type is the span of constant functio…

Let $\Theta_f$ be a theta block of $q$-order one that is a holomorphic Jacobi form of integral weight $k$ and integral index $N$ for $A_1$. Let $\operatorname{Grit}(\Theta_f)$ deno…

Let $m\equiv\pm3\pmod 8$, let $V={\rm I}_m\otimes\mathbb{Q}$ be the quadratic $\mathbb{Q}$-vector space from Case 3, and let $\mathbf{G}=\operatorname{SO}_V$. For…

Let $c0$ be an irreducible generic smooth representation of $cSO_{2n+1}(cQ_p)$ with conductor $p^m$. For $m\geq 0$, let $K(p^m)$ be the paramodular subgroup and let $J(p^m)$ be the…

Let $N$ be a positive integer, and let $\phi$ be a holomorphic Jacobi form whose Gritsenko lift is a Borcherds product. A pure theta block is a theta block whose defining function…

Let $B$ range over QM abelian fourfolds over $\mathbb{Q}$ of conductor $N^2$, and let $f$ range over cuspidal, nonlift Siegel paramodular newforms of genus $2$, weight $2$, and lev…

Let $p$) be a prime such that the space $S_3(\Gamma_p^{+})$ of antisymmetric paramodular cusp forms of weight $3$ is one dimensional. In this situation, the unique form gives a new…

Let $K(5)$ denote the paramodular group of level $5$, and let … be its graded ring of paramodular forms. Marschner's conjecture. The graded ring $M_*(K(5))$ is Cohen–Macaulay. The…

Let $N$ be square-free. Consider isogeny classes of abelian surfaces $A/{\mathbb Q}$ of conductor $N$ satisfying $\operatorname{End}_{\mathbb Q}A={\mathbb Z}$, and paramodular Sieg…

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 holomorphic paramodular cusp forms and holomorphic Siegel modular cusp forms be as in the source, and consider the local non-archimedean representations attached to the latter…

Let $N$ be squarefree, and let $F\in S^k(\Gamma^{\mathrm{para}}[N])$ be a Hecke eigenform that is not a Gritsenko lift. For fundamental discriminants $\ell$ and $D$ satisfying…

Twisted Böcherer conjecture.

Let $p$) be prime and let $F\in S^k(\Gamma^{\mathrm{para}}[p])^+$ be a paramodular Hecke eigenform of even weight $k$. For each fundamental discriminant $D<0$, let $A_F(D)$ denote…

Let $F\in S^k(\Gamma^{\text{para}}[p])^+$, where $S^k(\Gamma^{\text{para}}[p])^+$ is the plus space of paramodular cusp forms, and for a fundamental discriminant $D<0$ define … whe…

Let $f$ be a weight $2$ newform for the paramodular group $K(N)$, not in the span of the Gritsenko lifts. Let $k_f$ be the totally real number field generated by the Hecke eigenval…

Paramodular cohomology conjecture. The cuspidal cohomology $H^5_{\mathrm{cusp}}(\Gamma_0(N);\mathbb C)$ contains a $2d_h$-dimensional subspace spanned by Hecke eigenclasses. For ev…