Prime Determinant Matrices and the Sympletic Theta Function

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“…where Styer [9] shows that if A B C D ∈ Γ (n) , then there exists a symmetric, integral matrix T such that det(CT + D) = ±p for some arbitrarily large prime p. As in Styer [9] , we set Z * = Z − T, M * = M In T 0 In , and we observe that We apply Theorem 2 and find that…”

“…H. Stark [8] determines χ(M) in the important special case when both C and D are nonsingular and when pD −1 is integral for some odd prime p. R. Styer [9] extends Stark's results and includes the case where C is singular. We use the following theorem of [9] to compute the explicit theta multiplier of Θ F,H,ζ+,ζ− (Z, X). …”

Evaluating symplectic Gauss sums and Jacobi Symbols

“…They define a theta function involving a quadratic form of m variables, embed it into the symplectic theta function, and thus calculate the transformation formula when m is even. The author [11] has already extended their theorem to include m odd by using an analogy of Dirichlet's theorem concerning primes in an arithmetic progression. This is actually stronger than we need.…”

“…This is actually stronger than we need. Without reproducing the arguments of [1] and [11] The first factor on the right becomes Q Continuing this procedure with the obvious definitions and using Theorem 7b) to multiply the Jacobi symbols, we finally get The Gauss sum on the right can be evaluated using Theorems 3, 4, and 2 for any known (C, D), In particular, if άetD is odd squarefree, then the Gauss sum on the left equals (2 m det F/det D). Πpidetz?…”

“…This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]).…”