Theta Functions of Indefinite Quadratic Forms over Real Number Fields
نویسنده
چکیده
We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.
منابع مشابه
Theta functions of quadratic forms over imaginary quadratic fields
is a modular form of weight n/2 on Γ0(N), where N is the level of Q, i.e. NQ−1 is integral and NQ−1 has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly. Stark [8] gives a different proof by converting θ...
متن کاملTheta Functionswith Harmonic Coe⁄cients over Number Fields
is a modular form of weight n=2þ n on G0ðN Þ, where G 1⁄4 SL2ðZÞ and N is the level of Q, i.e., NQ 1 is integral and NQ 1 has even diagonal entries. This was proved by Schoeneberg [13] for even n and by Pfetzer [9] for odd n. Shimura [14] generalizes their results for arbitrary n and also computes the theta multiplier explicitly. Andrianov and Maloletkin [1, 2] generalize (1) and define theta s...
متن کاملMock modular forms and geometric theta functions for indefinite quadratic forms
Mock modular forms are central objects in the recent discoveries of new instances of Moonshine. In this paper, we discuss the construction of mixed mock modular forms via integrals of theta series associated to indefinite quadratic forms. In particular, in this geometric setting, we realize Zwegers’ mock theta functions of type ( p, 1) as line integrals in hyperbolic p-space.
متن کاملThe family of indefinite binary quadratic forms and elliptic curves over finite fields
In this paper, we consider some properties of the family of indefinite binary quadratic forms and elliptic curves. In the first section, we give some preliminaries from binary quadratic forms and elliptic curves. In the second section, we define a special family of indefinite forms Fi and then we obtain some properties of these forms. In the third section, we consider the number of rational poi...
متن کاملIndefinite Theta Series of Signature (1, 1) from the Point of View of Homological Mirror Symmetry
We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bund...
متن کامل