Manin ends his 1978 ICM talk with this remark:

I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic meaning. The first class to consider is that of the Schrödinger operators $-d^2/dx^2 + u(x)$ with algebro-geometric potentials $u(x)$ arising as solutions of the Korteweg-de Vries equation, for example: $u(x) = 2\wp(x)$ where $\wp$ is the Weierstrass function of an elliptic curve over $Q$. In fact, it seems that the values of this zeta-function at negative integers, which can be calculated explicitly, admit a $p$-adic interpolation.

Simply put, I'd be grateful to anyone who can explain to me what he's talking about. In particular, has "Gel'fand's suggestion" found explicit form, either as theorem or conjecture, anywhere in the literature?