Dimensions of paramodular forms and compact twist modular forms with involutions

arXiv2208.13578MaRDI QIDQ6506485

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Abstract: 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 $de{t}^{k}operatornameSym(j)$ with $kgeq3$ 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.

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