Acta Arithmetica (2000)

• Volume: 92, Issue: 1, page 1-9
• ISSN: 0065-1036

top

## Abstract

top
1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function ${\vartheta }_{Q}\left(z\right):={\sum }_{g\in {ℤ}^{n}}exp\pi i{}^{t}gQgz$, Im z > 0, is a modular form of weight n/2 on ${\Gamma }_{0}\left(N\right)$, where N is the level of Q, i.e. $N{Q}^{-1}$ is integral and $N{Q}^{-1}$ 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 ${\vartheta }_{Q}\left(z\right)$ into a symplectic theta function and then using the transformation formula for the symplectic theta function. In [4], we apply Stark’s method and use theta functions of indefinite quadratic forms to construct modular forms over totally real number fields. In this paper, we define theta functions attached to quadratic forms over imaginary quadratic fields. We show that these theta functions are modular forms of weight n/2 on some ${\Gamma }_{0}$ groups by regarding them as symplectic theta functions and then applying well known results for symplectic theta functions. In particular, the main result of [8] allows us to compute the theta multiplier for our theta functions in a very elegant way.

## How to cite

top

Olav K. Richter. "Theta functions of quadratic forms over imaginary quadratic fields." Acta Arithmetica 92.1 (2000): 1-9. <http://eudml.org/doc/207365>.

@article{OlavK2000,
abstract = {1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function $ϑ_Q(z) := ∑_\{g ∈ ℤ^n\} exp\{πi\{^tg\}Qgz\}$, Im z > 0, is a modular form of weight n/2 on $Γ_0(N)$, 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 $ϑ_Q(z)$ into a symplectic theta function and then using the transformation formula for the symplectic theta function. In [4], we apply Stark’s method and use theta functions of indefinite quadratic forms to construct modular forms over totally real number fields. In this paper, we define theta functions attached to quadratic forms over imaginary quadratic fields. We show that these theta functions are modular forms of weight n/2 on some $Γ_0$ groups by regarding them as symplectic theta functions and then applying well known results for symplectic theta functions. In particular, the main result of [8] allows us to compute the theta multiplier for our theta functions in a very elegant way.},
author = {Olav K. Richter},
journal = {Acta Arithmetica},
keywords = {quaternionic upper half-plane; theta function; quadratic forms},
language = {eng},
number = {1},
pages = {1-9},
url = {http://eudml.org/doc/207365},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Olav K. Richter
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 1
SP - 1
EP - 9
AB - 1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function $ϑ_Q(z) := ∑_{g ∈ ℤ^n} exp{πi{^tg}Qgz}$, Im z > 0, is a modular form of weight n/2 on $Γ_0(N)$, 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 $ϑ_Q(z)$ into a symplectic theta function and then using the transformation formula for the symplectic theta function. In [4], we apply Stark’s method and use theta functions of indefinite quadratic forms to construct modular forms over totally real number fields. In this paper, we define theta functions attached to quadratic forms over imaginary quadratic fields. We show that these theta functions are modular forms of weight n/2 on some $Γ_0$ groups by regarding them as symplectic theta functions and then applying well known results for symplectic theta functions. In particular, the main result of [8] allows us to compute the theta multiplier for our theta functions in a very elegant way.
LA - eng
KW - quaternionic upper half-plane; theta function; quadratic forms
UR - http://eudml.org/doc/207365
ER -

## References

top
1. [1] M. Eichler, Introduction to the Theory of Algebraic Numbers and Functions, Academic Press, New York, 1966.
2. [2] E. Hecke, Über die L-Funktionen und den Dirichletschen Primzahlsatz für einen beliebigen Zahlkörper, Nachr. Ges. Wiss. Göttingen Math.-phys. Kl. 1917, 299-318. Zbl46.0256.03
3. [3] W. Pfetzer, Die Wirkung der Modulsubstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Variablenzahl, Arch. Math. (Basel) 4 (1953), 448-454. Zbl0052.08703
4. [4] O. Richter, Theta functions of indefinite quadratic forms over real number fields, Proc. Amer. Math. Soc., to appear. Zbl1036.11016
5. [5] B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523. Zbl0020.20201
6. [6] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440-481. Zbl0266.10022
7. [7] C. Siegel, Indefinite quadratische Formen und Funktionentheorie II, Math. Ann. 124 (1952), 364-387. Zbl0046.27401
8. [8] H. Stark, On the transformation formula for the symplectic theta function and applications, J. Fac. Sci. Univ. Tokyo Sect. 1A 29 (1982), 1-12.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.