2017/10/25 by Gallenkämper, Jonas, Krieg, Aloys
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1710.09156
In this paper we consider the integral orthogonal group with respect to the quadratic form of signature (2,3) given by (\beginsmallmatrix 0 & 1 1 & 0 \endsmallmatrix) ⊥ (\beginsmallmatrix 0 & 1 1 & 0 \endsmallmatrix) ⊥ (-2N) for squarefree N∈ ℕ. The associated Hecke algebra is commutative and the tensor product of its primary components, which turn out to be polynomial rings over ℤ in 2 algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree 2 and level N, more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside Sp(2;ℚ) fails to be commutative if N> 1.