What connections are there between number theory and partial differential equations?

What connections are there between number theory and partial differential equations?

A belated partial/sample answer: first, to say that some function on a Euclidean or other space is a solution of a (natural!) PDE, perhaps the unique solution in a space of functions described by some integrability or other conditions, can be an excellent characterization of the thing. In many cases of traditional interest both in number theory and in physics, PDEs have many symmetries, and special solutions with symmetries often allow separation of variables reducing to ODEs, whose solutions have tractable asymptotics. Asymptotics are very handy for non-elementary functions.

Such things come up in number theory in examples such as the following. For context, Hecke observed that holomorphic binary theta series like $\theta(z)=\sum_{m,n} e^{\pi i z(m^2+n^2)}$ gives zeta functions of complex quadratic rings of algebraic integers. Prompted by Hecke, Maass looked for, and found, analogous automorphic forms (Maass' special waveforms) to produce zeta functions for real quadratic fields. These are eigenfunctions for the $SL_2(\mathbb R)$-invariant Laplacian $y^2(\partial_x^2+\partial_y^2)$ on the upper half-plane. At the time (1940s) this was a surprise, but with hindsight this has been assimilated pretty well.

In fact, the $L^2$ space on the quotient of the upper half-plane by $SL_2(\mathbb Z)$, with the invariant measure $dx\,dy/y^2$, decomposes in a very interesting way with respect to that Laplacian: there are genuine $L^2$ eigenfunctions consisting of cuspforms and also constants, and there is a 'continuous' part expressible as integrals ('wave packets') of (unhelpfully named "non-holomorphic") Eisenstein series $E_s$, the Eisenstein series being eigenfunctions of the Laplacian, but not $L^2$.

The return of number theory here is exemplified by the following particular thing. Even if one is directly interested only in holomorphic modular forms $f$ and associated $L$-functions, of course $|f|^2$ or $y^k\,|f|^2$ is no longer holomorphic, but/and the standard Rankin-Selberg integral expresses the (completed) $L$-function $L(s,f\times \bar{f})$ as the integral of $|f|^2$ against $E_s$. That is, that $L$-function has meaning in the spectral decomposition with respect to the Laplace-Beltrami operator, that it is the continuous-spectrum spectral coefficient!

Iwaniec' book "Spectral methods of automorphic forms" is a very accessible introduction to such things, and there are on-line discussions of such matters.

As in other answers somewhere on MSE or MO, it deserves to be added that "trace formulas" such as Selberg's can be viewed as studies of the resolvent of the Laplace-Beltrami operator (in rank-one groups, certainly).

While the Laplace-Beltrami operators relevant to automorphic forms are really "just" manifestations of the Casimir operator from the Lie algebra of the relevant Lie group, so that we could try to dismiss PDE talk as just concealed representation theory, that is a little misleading, insofar as there is some analysis that must be done, both local and global, and it is very handy to have an elliptic differential operator to be able to invoke elliptic regularity.

For that matter, both Harish-Chandra's discussion of repn theory and important developments such as Casselman's subrepn theorem (as in the Duke paper of Casselman and Milicic) make substantial use of systems of PDEs to characterize spherical functions and other stuff.

A different example is provided by Colin de Verdiere's proof of meromorphic continuation of Eisenstein series, by constructing a variant of that same Laplace-Beltrami operator, designed to have a compact resolvent, thus having a meromorphic continuation, from which the continuation of the Eisenstein series is obtained.