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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{zeta function of a Riemann surface} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry} [[!include complex geometry - contents]] \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra} [[!include AQFT and operator algebra contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{for_a_complex_torus__complex_elliptic_curve}{For a complex torus / complex elliptic curve}\dotfill \pageref*{for_a_complex_torus__complex_elliptic_curve} \linebreak \noindent\hyperlink{OfDiracOperatorTwistedByFlatConnection}{Of Dirac operators twisted by a flat connection}\dotfill \pageref*{OfDiracOperatorTwistedByFlatConnection} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{AnalogyWithArtinLFunction}{Analogy with Artin L-function}\dotfill \pageref*{AnalogyWithArtinLFunction} \linebreak \noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[zeta function]] naturally associated to a [[Riemann surface]]/[[complex curve]], hence the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on the Riemann surface (and hence hence essentially the [[Feynman propagator]] for the [[scalar fields]] on that surface) is directly analogous to the zeta functions associated with [[arithmetic curves]], notably the [[Artin L-functions]]. (\hyperlink{MinakshisundaramPleijel49}{Minakshisundaram-Pleijel 49}) considered the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on a [[Riemann surface]]. Motivated by the resemblance of the [[Selberg trace formula]] to Weil's formula for the sum of zeros of the [[Riemann zeta function]], (\hyperlink{Selberg56}{Selberg 56}) defined for any compact hyperbolic [[Riemann surface]] a [[zeta function]]-like expression, the \emph{[[Selberg zeta function]] of a Riemann surface}. (e.g. \hyperlink{Bump}{Bump, below theorem 19}). Much of this is more generally defined/considered on higher dimensional [[hyperbolic manifolds]]. That the Selberg zeta function is indeed proportional to the [[zeta function of an elliptic differential operator|zeta function]] of a [[Laplace operator]] is due to (\hyperlink{DHokerPhong86}{D'Hoker-Phong 86}, \hyperlink{Sarnak87}{Sarnak 87}), and that it is similarly related to the [[eta function of a self-adjoint operator|eta function]] of a [[Dirac operator]] on the given Riemann surface/hyperbolic manifold goes back to (\hyperlink{Milson78}{Milson 78}), with further development including (\hyperlink{Park01}{Park 01}). For review of the literature on this relation see also the beginning of (\hyperlink{Friedman06}{Friedman 06}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{for_a_complex_torus__complex_elliptic_curve}{}\subsubsection*{{For a complex torus / complex elliptic curve}}\label{for_a_complex_torus__complex_elliptic_curve} For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a [[complex torus]] (complex [[elliptic curve]]) equipped with its standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is \begin{displaymath} \zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,. \end{displaymath} The corresponding [[functional determinant]] is \begin{displaymath} \exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,, \end{displaymath} where $\eta$ is the [[Dedekind eta function]]. (recalled e.g. in \hyperlink{Todorov03}{Todorov 03, page 3}) \hypertarget{OfDiracOperatorTwistedByFlatConnection}{}\subsubsection*{{Of Dirac operators twisted by a flat connection}}\label{OfDiracOperatorTwistedByFlatConnection} For $A$ a [[flat connection]] on a [[Riemannian manifold]], write $D_A$ for the [[Dirac operator]] twisted by this connection. On a suitable [[hyperbolic manifold]], the [[partition function]]/[[theta function]] for $D_A$ appears in (\hyperlink{BunkeOlbrich94}{Bunke-Olbrich 94, prop. 6.3}) (and \hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, def. 3.1}) for the odd dimensional case). The corresponding Selberg zeta formula is (\hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, def. 4.1}). This has a form analogous to that of [[Artin L-functions]] with the flat connection replaced by a [[Galois representation]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{AnalogyWithArtinLFunction}{}\subsubsection*{{Analogy with Artin L-function}}\label{AnalogyWithArtinLFunction} That the Selberg/Ruelle zeta function is equivalently an [[Euler product]] of [[characteristic polynomials]] is due to (\hyperlink{Gangolli77}{Gangolli 77, (2.72)} \hyperlink{Fried86}{Fried 86, prop. 5}). That it is in particular the Euler product of characteristic polynomials of the [[monodromies]]/[[holonomies]] of the [[flat connection]] corresponding to the given [[group representation]] is (\hyperlink{BunkeOlbrich94}{Bunke-Olbrich 94, prop. 6.3}) for the even-dimensional case and (\hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a}) for the odd-dimensional case. Notice that this is analogous to the standard definition of an [[Artin L-function]] if one interprets a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the arithmetic fundamental group of an [[arithmetic curve]] and a [[Galois representation]] as a [[flat connection]]. \hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy} [[!include function field analogy -- table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} Original articles include \begin{itemize}% \item S. Minakshisundaram, ; \AA{} Pleijel, \emph{Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds} (1949), Canadian Journal of Mathematics 1: 242--256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145 (\href{http://cms.math.ca/10.4153/CJM-1949-021-5}{web}) \item [[Atle Selberg]], \emph{Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series}, Journal of the Indian Mathematical Society 20 (1956) 47-87. \item [[John Milson]], \emph{Closed geodesic and the $\eta$-invariant}, Ann. of Math., 108, (1978) 1-39 (\href{http://www.jstor.org/stable/1970928}{}) \end{itemize} Review includes \begin{itemize}% \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Selberg_zeta_function}{Selberg zeta function}} \item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Minakshisundaram–Pleijel_zeta_function}{Minakshisundaram--Pleijel zeta function}} \item [[Matthew Watkins]], citation collection on \emph{\href{http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm}{Selberg trace formula and zeta functions}} \item Bump, below theorem 19 in \emph{Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$} ([[BumpSpectralTheory.pdf:file]]) \end{itemize} Expression of the Selberg/Ruelle zeta function as an [[Euler product]] of [[characteristic polynomials]] is due to \begin{itemize}% \item Ramesh Gangolli, \emph{Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one}, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. (\href{http://projecteuclid.org/euclid.ijm/1256049498}{Euclid}) \item [[David Fried]], \emph{The zeta functions of Ruelle and Selberg. I}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 19 no. 4 (1986), p. 491-517 (\href{http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0}{Numdam}) \end{itemize} Discussion of the relation between, on the one hand, [[zeta function of an elliptic differential operator|zeta function]] of [[Laplace operators]]/[[eta function of a self-adjoint operator|eta funcstions]] of [[Dirac operators]] and, on the other hand, Selberg zeta functions includes \begin{itemize}% \item [[Eric D'Hoker]] [[Duong Phong]], \emph{Communications in Mathematical Physics}, Volume 104, Number 4 (1986), 537-545 (\href{http://projecteuclid.org/euclid.cmp/1104115166}{Euclid}) \item [[Peter Sarnak]], \emph{Determinants of Laplacians}, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. (\href{http://projecteuclid.org/euclid.cmp/1104159171}{Euclid}) \item [[Ulrich Bunke]], [[Martin Olbrich]], Andreas Juhl, \emph{The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function}, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405 \item [[Ulrich Bunke]], Martin Olbrich, \emph{Theta and zeta functions for locally symmetric spaces of rank one} (\href{http://arxiv.org/abs/dg-ga/9407013}{arXiv:dg-ga/9407013}) \end{itemize} and for odd-dimensional spaces also in \begin{itemize}% \item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one} (\href{http://arxiv.org/abs/dg-ga/9407012}{arXiv:dg-ga/9407012}) \item [[Ulrich Bunke]], [[Martin Olbrich]] \emph{$\Gamma$-Cohomology and the Selbeg zeta function} (\href{http://arxiv.org/abs/dg-ga/9411004}{arXiv:dg-ga/9411004}) \item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group} (\href{http://arxiv.org/abs/dg-ga/9603003}{arXiv:dg-ga/9603003}) \item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Selberg zeta and theta functions: a differential operator approach}, Akademie Verlag 1995 \item Jinsung Park, \emph{Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps} (\href{http://arxiv.org/abs/math/0111175}{arXiv:0111175}) \item [[Joshua Friedman]], \emph{The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations} (\href{http://arxiv.org/abs/math/0410067}{arXiv:math/0410067}) \item [[Joshua Friedman]], \emph{Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations}, Communications in Mathematical Physics (\href{http://arxiv.org/abs/math/0605288}{arXiv:math/0605288}) \end{itemize} See also \begin{itemize}% \item [[Andrey Todorov]], \emph{The analogue of the Dedekind eta function for CY threefolds}, 2003 \href{http://www.ma.huji.ac.il/conf/crelle.pdf}{pdf} \end{itemize} [[!redirects Selberg zeta function of a Riemann surface]] \end{document}