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The prime geodesic theorem (of Margulis?) states that on a compact surface of (constant?) negative curvature, the number of prime closed geodesics of length at most $L = \log x$ is approximately $e^L/L = x/\log x$ as $x$ grows. This is commonly viewed as an analogue of the prime number theorem.

Reading the report on Mirzakhani's work in relation to her receiving the Fields medal, I learned that she proved that the number of simple (i.e. non-self-intersecting) prime closed geodesics of length at most $L$ is of the order of $L^{6g-6}$, where $g$ is the genus of the surface, as $L$ grows. What would be the analogue of simple prime closed geodesic in the context of prime numbers and what would one expect the analogue of Mirzakhani's result to be?

Edit: My colleague Alan Reid suggested to look at the universal cover to distinguish simple from non-simple closed geodesics. Since I want to keep the analogy with arithmetic, instead it's better to look at finite covers. Is it fair to say that simple closed geodesics are more likely to become non-prime in finite covers? That would suggest that "simple" primes would be those more likely to split in finite extensions.

Edit 2: My history is sloppy. See GH's comment and Igor's answer.

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  • $\begingroup$ Mirzakhani's result concerns hyperbolic (constant negative curvature) surfaces of finite area, i.e. quotients of the upper half-plane by a discrete subgroup of isometries. In this context, the prime geodesic theorem is essentially due to Selberg, e.g. the analogue of the Riemann zeta is the Selberg zeta. $\endgroup$ – GH from MO Aug 13 '14 at 3:33
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    $\begingroup$ Birman and Series (J. Lond Math Soc 1984) characterize non-self-intersection of a closed loop via a group-theoretic property of the corresponding element in the fundamental group. Their characterization depends on the Nielesen generators available for $\pi_1$ of cpt orientable surfaces with non-empty bdry. Wild out of the blue idea: try something similar, with Frobenius at $p$, and the usual generators of the Grothendieck-Teichmuller group. $\endgroup$ – Marty Aug 13 '14 at 5:12
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    $\begingroup$ What would the analog be for the exponent $6g-6$ (which arises as the dimension of Teichmuller space - or the space of measured laminations)? If it is $\asymp \log{|D_{K/\mathbb{Q}}|}$, then a corresponding asymptotic for such "simple prime ideals" in a number field $K$ would be of the shape $|D_{K/\mathbb{Q}}|^{C\log{\log{x}}}$, which would be rather strange. $\endgroup$ – Vesselin Dimitrov Aug 13 '14 at 8:29
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First of all, the prime geodesic theorem is due to Sarnak, in Margulis' result the geodesics need not be prime. Secondly, for constant curvature, the result goes back to Huber and Selberg, with precise error terms. Margulis extended this to variable negative curvature (with no error term), and that was sharpened (at least theoretically) by Dolgopyat to show that there is an exponentially decaying error term, but the exponent is not explicit at all.

Thirdly, the order of growth result for simple geodesics is due not to Maryam, but to yours truly:

%0 Journal Article %A Rivin, Igor %T Simple curves on surfaces %J Geom. Dedicata %V 87 %D 2001 %N 1-3 %P 345--360 %@ 0046-5755 %L MR1866856 (2003c:57018) %R doi:10.1023/A:1012010721583 %U http://dx.doi.org/10.1023/A:1012010721583

Maryam proved an asymptotic result, which (as is not surprising to number theorists) is weaker in practice (my result has explicit bounds, hers does not), but certainly conceptually better (with modern technology - mixing of Teichmuller geodesic flow - one can presumably show the existence of error term a la Dolgopyat, but nothing explicit).

Lastly, and getting back to the actual question, the connection between prime numbers and closed geodesics goes via the Selberg zeta function, and its analogy to the Riemann zeta function. So, I will not speak of prime numbers, but the lattice point counting. It turns out that the counting of all closed geodesics and simple closed geodesics is conceptually very similar, despite completely different result that one gets: counting closed geodesics corresponds to counting points of an orbit of a Fuchsian group (for example, $PSL(2, Z)$) in a disk in the hyperbolic plane (that was what Selberg counted initially). The insight is that simple closed curves lie in a finite set of orbits of the mapping class group, and so counting simple closed geodesic reduces to lattice point counting for the mapping class group on Teichmuller space. I am not sure what the number-theoretic analogue would be (that is, arithmetic geometers certainly study moduli, but I am not sure what easy-to-understand things this counts). Notice that the way Huber/Selberg count lattice points in symmetric spaces is via harmonic analysis, and so what is needed for a parallel theory for Teichmuller space is developing tools for analysis on Teichmuller/moduli spaces. Mirzakhani made the first step in this, by combining Greg McShane's amazing identity (or an extension thereof) with a simple idea of Siegel. This allowed her to integrate the constant function. A baby step, to be sure (an anecdote: I applied for an NSF grant once to develop analysis on Teichmuller space. It was turned down as "too narrow").

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    $\begingroup$ Thanks, that's very helpful and informative. I'm still in the dark with regards to my question. Marty suggested in the comments to look at the fundamental group but I don't understand what is the criterion of Birman and Series. For number fields, the absolute Galois group is too big and the analogue of the fundamental group (the Galois gp of the max unramified extension) is too variable and sometimes too small. Maybe the thing to do, from what you say, is to look first at function fields over finite fields. There one has fundamental groups and moduli spaces and perhaps a mapping class group. $\endgroup$ – Felipe Voloch Aug 14 '14 at 23:54

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