Textbooks talk at length of the modular properties of $\theta(z)$ or $\tau(z)$ and the prominent role of $SL(2,\mathbb{Z})$ or one of the congruence groups.

In that case, aren't the basic objects in elementary number theory such as the set of squares $\square = \{ n^2 : n \in \mathbb{Z}\}$ also related to the geometry of $\mathbb{H}/\Gamma_0(4)$ ?

Texts on modular forms say nothing about continued fractions, symbolic coding or the geodesic flow on the modular surface. Am I missing something here?

  • $\begingroup$ I know I should be cautious with such open-ended questions. Not an expert that I could make this question more sharp. $\endgroup$ – john mangual Mar 18 '16 at 16:02
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    $\begingroup$ I have to disagree with the premise that "texts on modular forms say nothing about..." Probably the difficulty is too narrow a selection of "texts". After all, Gelfand et al's work (1950s) on ergodic properties of geodesic and horocyclic flows led to his defn of "cuspform". Ratner's theorem has had many applications in the last several years. The connection with continued fractions is 100+ years old. What is the real question? $\endgroup$ – paul garrett Mar 18 '16 at 21:36
  • $\begingroup$ I don't see much about continued fractions in Apostol or the texts of Goro Shimura. How is Ratner theorem related to modular forms?? Or the horocycle flow?? $\endgroup$ – john mangual Mar 18 '16 at 22:15
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    $\begingroup$ Apostol's text is entry-level, and is decades old. Shimura's books have very specific goals, and that most-elementary one has very limited goals. These are not broad samples of what people know, etc. The book by Einsiedler and Ward "Ergodic theory with a view toward Number Theory" tells about many connections of ergodic theory to number theory, including an introduction to automorphic examples. $\endgroup$ – paul garrett Mar 18 '16 at 22:45
  • $\begingroup$ Einsiedler and Ward don't say much about automorphic forms. Instead they write of $L^2(X)$ which is space of modular form $\endgroup$ – john mangual Mar 18 '16 at 23:06

Yes, this is very open-ended. A 'big picture' answer to your question is that the closed geodesics on the surface, and the eigenvalues of the Laplacian (corresponding to Maass forms) are on opposite sides of the Selberg trace formula. The geodesics are analogous to primes, and the eigenvalues analogous to the Riemann zeros. The Selberg trace formula is the analogy of Riemann's explicit formula.

Of course, there's lots of other possible answers...

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  • $\begingroup$ Just curious, what other possible answers do you have in mind? $\endgroup$ – Pig Mar 18 '16 at 21:33

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