Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations

Abstract

Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let \(\chi \) denote a finite dimensional unitary representation of the fundamental group of M. Let \(\Delta \) denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with \(\chi \). From the spectral theory of \(\Delta \), there are three distinct sequences of numbers: the first coming from the eigenvalues of \(L^{2}\) eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, \(\mathcal {Z}_-(s,z)\) and \(\mathcal {Z}_+(s,z)\) that encode the spectrum of \(\Delta \) in such a way that they can be used to define the regularized determinant of \(\Delta -z(1-z)I\). The resulting formula for the regularized determinant of \(\Delta -z(1-z)I\) in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry \(z\leftrightarrow 1-z\).

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Notes

  1. 1.

    The scattering determinant \(\phi (s)\) is actually real valued on \({\mathbb {R}}\) and it follows that \(d(n) \in {\mathbb {R}}.\)

  2. 2.

    Note that our notation is very different from his. Note that his k represents the weight of his forms, which we take as zero in our paper.

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Correspondence to Jay Jorgenson.

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J. S. Friedman, The views expressed in this article are the author’s own and not those of the U.S. Merchant Marine Academy, the Maritime Administration, the Department of Transportation, or the United States government. Support for the second-named author was provided by several PSC-CUNY Awards, which are jointly funded by The Professional Staff Congress and The City University of New York.

A Alternate expressions

A Alternate expressions

The multiplicities \(m_n\) of the trivial zeros of the Selberg zeta function carry important information related to the surface M and the character \(\chi \). For this reason, we will deduce a different expression for \(m_n\) (see Eq. (A.6)) from which it will be obvious that \(m_n\) are nonnegative integers. Moreover, we will construct a different order-two entire function \({{\tilde{G}}}_1(s)\), given in terms of the gamma and the Barnes double gamma function (see Eq. (A.7)) and such that its null set coincides with the set of negative integers \(-n\) with multiplicities \(m_n\).

A.1 An alternate expression for the multiplicities \(m_n\)

We simplify (2.12) by combining it with (2.1) multiplied by \(\frac{h}{2\pi }\) and (3.3) to get

$$\begin{aligned} m_n= & {} h\left( 2g-2 + \mathbf{c} + \mathbf{e} \right) (2n+1) \nonumber \\&- \sum _{j=1}^h \sum _{\{R\}_{\Gamma }} \frac{1}{d_R} \left( 2n+1 + \sum _{k=1}^{d_R-1} \left( \omega (R)^k_j \right) \frac{\sin \left( \frac{k\pi (2n+1)}{d_R}\right) }{\sin \left( \frac{k\pi }{d_R}\right) } \right) \end{aligned}$$
(A.1)

By (A.1), we can focus on the case of unitary characters rather than unitary representations.

Lemma A.1

Let \(\omega \) be a unitary character of the finite cyclic group \(\left\langle R \right\rangle \), where R is elliptic of order d. Further let \(\omega (R)=\exp (2 \pi i q/d)\) for some integer q,  with \(0 \le q \le d-1.\) Let \(n \in \mathbb {N}= \{0,1,2,\dots \},\) then

$$\begin{aligned} \frac{1}{d}\left( 2n+1 + \sum _{k=1}^{d-1}\frac{\omega ^k(R)}{\sin \left( \frac{k\pi }{d}\right) }\sin \left( \frac{k\pi (2n+1)}{d}\right) \right) = \left| \{t \in \mathbb {Z}~|~ td \in \{-n+q,\dots n+q\} \} \right| , \end{aligned}$$
(A.2)

where |A| denotes the cardinality of the finite set A.

Proof

We apply the identity

$$\begin{aligned} \frac{\sin (k\pi (2n+1)/d)}{\sin (k\pi /d)}= \sum _{j=-n}^{n}\exp (2\pi i k j/d) \end{aligned}$$

and obtain

$$\begin{aligned}&\frac{1}{d}\left( 2n+1 + \sum _{k=1}^{d-1}\frac{\omega ^k(R)}{\sin \left( \frac{k\pi }{d}\right) }\sin \left( \frac{k\pi (2n+1)}{d}\right) \right) \nonumber \\&\quad = \frac{1}{d}(2n+1) + \sum _{k=1}^{d-1} \frac{1}{d} \exp (2 \pi i k q/d)\sum _{j=-n}^{n}\exp (2\pi i k j/d) \nonumber \\&\quad = \frac{1}{d}(2n+1) + \frac{1}{d} \sum _{j=-n}^{n} \sum _{k=1}^{d-1} \exp (2 \pi i kq/d) \exp (2\pi i k j/d) \nonumber \\&\quad = \frac{1}{d} \sum _{j=-n}^{n} \sum _{k=0}^{d-1} \exp (2 \pi i k (q+j)/d) \end{aligned}$$
(A.3)

The inner sum on the right is equal to d iff \(d | (q+j).\) As j runs from \(-n,-n+1,\dots ,n,\) the number of such j is equal to the number of multiples of d in the integer range \(-n+q,\dots ,n+q.\) Recalling the factor \(\tfrac{1}{d}\) out in front proves the lemma. \(\square \)

Lemma A.2

Let dqn be integers with \(2 \le d, 0 \le q \le d-1,\) and \(0 \le n.\)

Then,

$$\begin{aligned} \left| \{t \in \mathbb {Z}~|~ td \in \{-n+q,\dots n+q\} \} \right| = \lfloor \frac{n+q}{d} \rfloor + \lfloor \frac{n+d-q}{d} \rfloor \end{aligned}$$

Here, \(\lfloor x\rfloor \) denotes the floor function.

Proof

For \(a \le b,\) both integers, define \(f(a,b,d) = \left| \{t \in \mathbb {Z}~|~ td \in \{a,\dots b\} \} \right| \)

Now, let ab be integers with \(a < 0\) and and \(b > 0.\) Then,

$$\begin{aligned} f(a,b,d)= & {} f(-a,0,d) + f(0,b,d) - 1 = f(0,-a,d) + f(0,b,d) + 1\nonumber \\= & {} \lfloor \frac{d-a}{d} \rfloor + \lfloor \frac{b}{d} \rfloor \end{aligned}$$
(A.4)

To prove the lemma, we consider the following cases. The first case, \(a = -n+q < 0,\) and \(b=n+q > 0\) follows from (A.4).

In the case when \(n=0,\) the lemma is trivial.

Next, consider the case where \(0 < -n + q.\) Since \(q < d,\) and \(n \ge 0,\) it follows that \(0<-n+q < d.\) If \(n+q < d,\) then it follows that

$$\begin{aligned} f(-n+q,n+q,d) = 0 = \lfloor \frac{n+q}{d} \rfloor + \lfloor \frac{d + (n-q)}{d} \rfloor = 0 + 0, \end{aligned}$$

and the lemma is verified in this case.

Finally, we are left with the case \(0<-n+q < d,\) and \(d < n+q.\) We shift the integer interval \(\{-n+q,\dots n+q\}\) to the left by d,  and obtain

$$\begin{aligned} \left| \{t \in \mathbb {Z}~|~ td \in \{-n+q,\dots n+q\} \} \right| = \left| \{t \in \mathbb {Z}~|~ td \in \{-n+q-d,\dots n+q-d\} \} \right| . \end{aligned}$$

We can apply (A.4) to the shifted interval, and we obtain

$$\begin{aligned} f(-n+q-d,n+q-d,d )= & {} \lfloor \frac{d-(-n+q-d)}{d} \rfloor + \lfloor \frac{n+q-d}{d} \rfloor \nonumber \\= & {} \lfloor 1 + \frac{n+d-q}{d} \rfloor + \lfloor -1 + \frac{n+q}{d} \rfloor \nonumber \\= & {} \lfloor \frac{n+d-q}{d} \rfloor + \lfloor \frac{n+q}{d} \rfloor , \end{aligned}$$
(A.5)

where the last equality follows because both \(n+q\) and \(n+d-q\) are positive. \(\square \)

One should note that Lemma A.2 is false for arbitrary nqd.

Combining (A.1), Lemma A.1, and Lemma A.2 we arrive at the following alternate expression for \(m_n\):

$$\begin{aligned} m_n= & {} h\left( 2g-2 + \mathbf{c} + \mathbf{e} \right) (2n+1) \nonumber \\&- \sum _{\{R\}_{\Gamma }} \sum _{j=1}^h \left( \lfloor \frac{n+q(R)_j}{d_R} \rfloor + \lfloor \frac{n+d_R-q(R)_j}{d_R} \rfloor \right) \end{aligned}$$
(A.6)

A. 2 Double gamma function representation of the trivial zeros of the Selberg-zeta function

Recall that the Barnes double Gamma function, which is an entire order-two function defined by (2.4), has a zero of multiplicity n,  at each point \(-n \in \{-1,-2,\dots \}.\) We start with the following lemma.

Lemma A.3

Let dqn be integers with \(2 \le d, 0 \le q \le d-1,\) and \(0 \le n.\) Set

$$\begin{aligned} g(n,q,d) = \lfloor \frac{n+q}{d} \rfloor + \lfloor \frac{n+d-q}{d} \rfloor . \end{aligned}$$

For \(s \in \mathbb {C},\) define

$$\begin{aligned} G_{q,d}(s) = \prod _{m=0}^{d-1} G\left( \frac{s-q+m}{d} + 1 \right) G\left( \frac{s-(d-q)+m}{d} + 1 \right) . \end{aligned}$$

Then, \(G_{q,d}(s)\) is an entire function with the set of zeros being the set of negative integers \(\{-n:n = 1,2,\dots \}\) and each zero \(s =-n\), \(n = 1,2,\dots \) has multiplicity g(nqd).

Proof

We study \(G_{q,d}(s)\) at \(s = -n.\) Since \(m \in \{0,\dots ,d-1\},\) the number \(\frac{-n-q+m}{d}\) is a negative integer for exactly one value, say \(m = m_1,\) in which case

$$\begin{aligned} \lfloor \frac{-n -q +m_1}{d} \rfloor = -\lfloor \frac{n+q}{d} \rfloor . \end{aligned}$$

Therefore, \(\prod _{m=0}^{d-1} G\left( \frac{s-q+m}{d}+1\right) \) has a zero of order \(\lfloor \frac{n+q}{d} \rfloor \) at \(s = -n\) for any positive integer n. Similarly, \(\prod _{m=0}^{d-1}G\left( \frac{s-(d-q)+m}{d} + 1 \right) \) has a zero of order \(\lfloor \frac{n+q-d}{d} \rfloor \) at \(s = -n.\)

Moreover, \(\frac{s-q+m}{d} = -k \) for some \(k = 1,2,3,\ldots ,\) iff \(s = -kd + q - m\) is a negative integer. This shows that there are no zeros of \(\prod _{m=0}^{d-1} G\left( \frac{s-q+m}{d}+1\right) \) different from negative integers. A similar conclusion holds for \(\prod _{m=0}^{d-1}G\left( \frac{s-(d-q)+m}{d} + 1 \right) \) and the proof is complete. \(\square \)

Recall that \(\{R\}_{\Gamma }\) are the classes of elliptic elements of \(\Gamma \) and that there are \(\mathbf{e} \) of them. Further, recall the notation \(\mathbf{c} ,h.\) Recall that \(\omega (R)_j,\) for \(j=1\dots h\) are the eigenvalues (and \(d_R\mathrm {th}\)-roots of unity) of \(\chi (R) \), and that \(q(R)_j,\) with \(0 \le q(R)_j \le d_R-1\) is defined by (3.4).

Definition A.4

With the notation above, we define

$$\begin{aligned} G_E(s) = \prod _{\{R\}_{\Gamma }} \prod _{j=1}^{h} \prod _{m=0}^{d_R - 1} G\left( \frac{s-q(R)_j+m}{d_R} + 1 \right) G\left( \frac{s-\left( d_R-q(R)_j\right) +m}{d_R} + 1 \right) . \end{aligned}$$

and

$$\begin{aligned} {{\tilde{G}}}_1(s)= \left( G_E(s) \right) ^{-1}\left( \frac{(2\pi )^{-s} (G(s+1))^2}{\Gamma (s)}\right) ^{h(2g-2+\mathbf{c} + \mathbf{e} )} \end{aligned}$$
(A.7)

Lemma A.4

The function \({{\tilde{G}}}_{1}(s)\) is an entire of order two with zeros at points \(-n \in -\mathbb {N}\) and corresponding multiplicities \(m_n.\)

Proof

The function \(\left( \frac{(2\pi )^{-s} (G(s+1))^2}{\Gamma (s)}\right) ^{h(2g-2+\mathbf{c} + \mathbf{e} )}\) possesses zeros at points \(s=-n\) with multiplicity \(h(2g-2+\mathbf{c} + \mathbf{e} )(2n+1)\); hence, the statement follows by combining Eq. (A.1) with Lemmas A.1A.2 and A.3. \(\square \)

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Friedman, J.S., Jorgenson, J. & Smajlović, L. Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations. Lett Math Phys 111, 15 (2021). https://doi.org/10.1007/s11005-021-01357-6

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Keywords

  • zeta regularization
  • determinant of the Laplacian

Mathematics Subject Classification

  • 11M36
  • 11F72