Paramodular conjecture (awaiting review)

The paramodular conjecture of Brumer and Kramer [arXiv:1004.4699] makes precise an earlier conjecture of Yoshida [eudml.org/doc/142747]. The conjecture states that for every abelian surface $A/\Q$ of paramodular type (meaning $\End(A)=\Z$) there exists a corresponding paramodular newform of level $N$ equal to the conductor of $A$.

For squarefree integers $N$ it is further conjectured that this correspondence gives a bijection between isogeny classes of paramodular type abelian surfaces of conductor $N$ and paramodular newforms of level $N$.