In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like primes in a number field.

That is, given a finite normal (i.e. ``Galois'') cover $M\to N$ of Riemannian manifolds, the preimage of prime geodesic $\mathfrak{p}$ in $N$ is a disjoint union $\mathfrak{P}_1,...,\mathfrak{P}_k$ of prime geodesics in $M$. The degree $f_i$ of the covering map $\mathfrak{P}_i\to \mathfrak{p}$ of circles should be thought of as the inertia degree of $\mathfrak{P}_i/\mathfrak{p}$, and the Frobenius $\sigma_{\mathfrak{P}_i}\in \text{Deck}(M/N)$ acts on $\mathfrak{P}_i$ by a $1/f_i$ rotation. Note $[M:N]=\sum f_i$. Other similar properties also hold, for instance there is also a Chebatorev density theorem.

My question is: is there an analogue of unique factorisation of ideals in this context?

I don't know what the right analogue of a non-prime ideal is here: perhaps $\mathfrak{p}^2$ should be $\mathfrak{p}$, but the geodesic loops round twice, but I can't think of the right definition for $I$ in general.


These questions are answered in Darin Brown's very nice paper:

Brown, Darin, Lifting properties of prime geodesics, Rocky Mt. J. Math. 39, No. 2, 437-454 (2009). ZBL1170.53022.

(available free).

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  • $\begingroup$ Thank you. For those not wanting to look through the paper: I think the paper hasn't found any geometric object corresponding to ideals, instead talking in terms of the formal product of prime ideals. The exception to this is prime power ideals, which correspond to prime geodesics looped round $n$ times. $\endgroup$ – Meow Jul 28 '18 at 18:28

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