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Theta functions of quadratic forms over imaginary quadratic fields

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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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
92
Numer
1
Strony
1-9
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-02-15
poprawiono
1999-09-06
Twórcy
  • Department of Mathematics, University of California at Santa Cruz, Santa Cruz, CA 95064, U.S.A.
Bibliografia
  • [1] M. Eichler, Introduction to the Theory of Algebraic Numbers and Functions, Academic Press, New York, 1966.
  • [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.
  • [3] W. Pfetzer, Die Wirkung der Modulsubstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Variablenzahl, Arch. Math. (Basel) 4 (1953), 448-454.
  • [4] O. Richter, Theta functions of indefinite quadratic forms over real number fields, Proc. Amer. Math. Soc., to appear.
  • [5] B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523.
  • [6] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440-481.
  • [7] C. Siegel, Indefinite quadratische Formen und Funktionentheorie II, Math. Ann. 124 (1952), 364-387.
  • [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.
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