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Title Indefinite Theta Functions and Zeta Functions
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher University of Michigan
Abstract We define an indefinite theta function in dimension g and index 1 whose modular parameter transforms by a symplectic group, generalizing a construction of Sander Zwegers used in the theory of mock modular forms. We introduce the indefinite zeta function, defined from the indefinite theta function using a Mellin transform, and prove its analytic continuation and functional equation. We express certain zeta functions attached to ray ideal classes of real quadratic fields as indefinite zeta functions (up to gamma factors). A Kronecker limit formula for the indefinite zeta function – and by corollary, for real quadratic fields – is obtained at s=1. Finally, we discuss two applications related to Hilbert's 12th problem: numerical computation of Stark units in the rank 1 real quadratic case, and computation of fiducial vectors of Heisenberg SIC-POVMs.
Subjects/Keywords number theory; indefinite theta function; zeta function; real quadratic field; Kronecker limit formula; SIC-POVM; Mathematics; Science
Contributors Lagarias, Jeffrey C (committee member); Doering, Charles R (committee member); Koch, Sarah Colleen (committee member); Prasanna, Kartik (committee member); Snowden, Andrew (committee member); Zieve, Michael E (committee member)
Language en
Rights Unrestricted
Country of Publication us
Record ID handle:2027.42/140957
Repository umich
Date Retrieved
Date Indexed 2019-08-21
Grantor University of Michigan, Horace H. Rackham School of Graduate Studies
Issued Date 2017-01-01 00:00:00
Note [thesisdegreename] PHD; [thesisdegreediscipline] Mathematics; [thesisdegreegrantor] University of Michigan, Horace H. Rackham School of Graduate Studies; [bitstreamurl] https://deepblue.lib.umich.edu/bitstream/2027.42/140957/1/gkopp_1.pdf;

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…symmetric matrix whose imaginary (1) part has signature (g − 1, 1); that is, Ω ∈ Hg . Define the indefinite theta function   c2 > X c Im(Ωn + z) 1 c1 ,c2 > >  (1.16) Θ [f ](z, Ω) = Ef  q n Ωn…

…The definite (Riemann) theta function is, for z ∈ Cg and Ω ∈ Hg , X 1 > > n Ωn + n z . (2.2) Θ(z; Ω) = e 2 n∈Zg Definition II.3. When g = 1, the definite theta functions is called a Jacobi theta function and is denoted by…

…functions Proposition II.8. The definite theta function for z ∈ Cg and Ω ∈ Hg satisfies the following transformation law with respect to the z variable, for a + Ωb ∈ Zg + ΩZg : (2.7) 1 > > Θ(z + a + Ωb, Ω) = e − b Ωb − b z Θ(z, Ω…

…x29; is an imaginary quadratic field, the modular function will evaluate to an algebraic number in a (generally non-abelian) extension of Q(z). 1.1.3 From indefinite theta functions to a new Kronecker limit formula We present a…

…the modular parameter of an indefinite theta function lives (1) (k) is Hg , where Hg is defined as follows. Definition II.14. For 0 ≤ k ≤ g, we define the Siegel intermediate half-space of genus g and index k to be (1.12)…

…x29;. 1.2.3 Indefinite theta functions and indefinite theta nulls with characteristics The incomplete Gaussian transform provides variable coefficients used to define an indefinite theta function. Definition II.22. Let Ω = N +iM be a complex…

…not both real, also assume that f is holomorphic. Set Θc1 ,c2 (z, Ω) := Θc1 ,c2 [1](z, Ω). Zwegers’s theta function is defined in arbitrary dimension g for real cj when N is a scalar multiple of M . More precisely, if M is…

…using a Mellin transform of the indefinite theta function with characteristics. (1) Definition III.2. Let Ω = N + iM ∈ Hg . The indefinite zeta function is (1.20) c1 ,c2 ζ̂p,q (Ω, s) Z ∞ = 0 1 ,c2 Θcp,q (tΩ)…