Cunningham, CliftonGreenberg, MatthewShahabi, Majid2018-09-202018-09-202018-09-13Shahabi, M. (2018). Smooth Integral Models for Certain Congruence Subgroups of Odd Spin Groups (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/32950http://hdl.handle.net/1880/107787In this thesis, we introduce a family of congruence subgroups for general odd spin groups GSpin(2n+1)/Q (see Chapter 1 for definition of GSpin(2n+1)). We prove that our congruence subgroups for GSpin(2n+1) admit integral models that are smooth group schemes over Z with generic fibre isomorphic to GSpin(2n+1)/Q (Proposition 2.7 and Theorem 2.9), have specific special fibres (Proposition 2.12 and Theorem 2.13), and generalize Hecke's congruence subgroup for GL2(Q) (Proposition 2.15) and the paramodular subgroup for GSp4(Q) (Proposition 2.16), where N is a fixed positive integer. We also study a family of congruence subgroups for special odd orthogonal groups SO(2n+1)/Q, introduced by Gross and Tsai [G2, T1, T2]. These congruence subgroups are expected to appear in anologs of modularity for abelian varieties predicted by the Langlands program [G2,CD]. Using scheme theory, we prove that these congruence subgroups for SO(2n+1) admit integral models that are smooth group schemes over Z[1/2] with generic fibre isomorphic to SO(2n+1)/Q (Theorems 3.6, 3.16, 3.10, 3.17), have specific special fibres (Propositions 3.11 and 3.18, and Theorems 3.12 and 3.19), and have Zp-points isomorphic to the congruence subgroups of SO(2n+1) defined by Gross and Tsai [G2, T1, T2] (Theorems 3.13 and 3.20). We also prove that there exists a close relationship between our integral models for these congruence subgroups of GSpin(2n+1) and SO(2n+1) (Theorem 4.1).engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Smooth Group SchemeIntegral ModelCongruence SubgroupOdd Spin GroupsMathematicsPhysicsStatisticsdoctoral thesishttp://dx.doi.org/10.11575/PRISM/32950