Incorporating the zonal spherical function (zsf) problems on real and p‐adic hyperbolic planes into a Zakharov–Shabat integrable system setting, we find a wide class of integrable evolutions that respect the number‐theoretic properties of the zsf problem. This means that at all times these real and p‐adic systems can be unified into an adelic system with an S matrix that involves (Dirichlet, Langlands, Shimura,...) L functions.

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© 1995 American Institute of Physics.
1995
American Institute of Physics
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