Brumer–Kramer Paramodular Conjecture

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

$$K(N)=\gamma M_4(\mathbf{Z})\gamma^{-1}\cap\operatorname{Sp}_4(\mathbf{Q}),\qquad \gamma=\operatorname{diag}[1,1,1,N].$$

A weight-$2$ Siegel modular form $F$ is required not to lie in the space spanned by the Saito–Kurokawa lifts, to have level $K(N)$, and to have rational eigenvalues; forms are considered up to scalar multiplication. Brumer–Kramer Paramodular Conjecture. There is a one-to-one correspondence between these isogeny classes and such forms $F$. Moreover, the $L$-series of $A$ and $F$ should agree, and for every prime $p$ not dividing $N$, the $p$-adic representation

$$V_p(A):=T_p(A)\otimes\mathbf{Q}_p$$

of $A$ should be isomorphic to the representation associated to $F$, where $T_p(A)$ is the $p$-adic Tate module. This conjecture refines Yoshida's correspondence by specifying the paramodular level and the associated arithmetic data; the paper applies deformation-theoretic methods to establish modularity in particular cases, while the full correspondence remains open.