Dimensions of paramodular forms and compact twist modular forms with involutions
The dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus has been known by our previous works. Here we give a new dimension formula for those with plus or minus sign of the Atkin-Lehner involution for the prime level case. Our formula is based on a class number formula of some quinary lattices by Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. By a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornar\'ia, the algebraic modular forms of the above type with signs have a good correspondence with paramodular forms with signs. So together with our results above, we also give a dimension formula for paramodular forms of degree two of prime level of any weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$ with plus or minus sign of the Atkin-Lehner involution. In particular we give the complete list of primes $p$ such that there is no paramodular cusp form of level $p$ of weight 3 with plus sign. It is known that this has a geometric meaning on the moduli of Kummer surfaces associated to $(1,p)$ polarizations.