## Theta Functions and Kubota Homomorphisms for the Symplectic Group Over the Gaussian Integers |

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### Contents

n Symplectic Theta Functions and the Pliicker Coordinates | 4 |

Sp2n Qj and the Quaternionic Upper Half Space | 24 |

The Theta Function on Sp2n Q0 and Its Transformation Formula | 33 |

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### Common terms and phrases

2n-ir+l Af(i C-lD Chapter characteristic polynomial choices of square commutes congruence subgroup coprime corresponding matrix corresponding symmetric space defined Det(CZ Dirichlet's theorem eighth root explicit formula Friedberg 9 Friedberg and Hoffstein Gauss sums GAUSSIAN INTEGERS give a formula give an explicit given by analytic Hence Hermitian and positive homomorphisms of Kubota ideal of norm integral invertible matrix Kubota 15 KUBOTA HOMOMORPHISMS l)th column L*+i LEMMA Milnor and Serre minor modP n x n Note odd prime Ozlem Erverdi Imamoglu parametrize the coset Pliicker Plucker coordinates positive definite prime ideal primes in progressions Proof PROPOSITION 3.7 prove the transformation quadratic residue symbol quadratic symbol quaternionic upper half real and positive root of det(C root of unity Siegel modular form Siegel upper half signature of b-1C square root symmetric matrix symplectic group Sp(2n symplectic matrix symplectic theta function THEOREM 2.7 theorem on primes transformation formula upper half space zero