I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism.

... can anyone justify this claim?

Given a von Neumann algebra $M$, its

modular flowis a canonically defined homomorphism $$ \mathbf{\Phi}: i\mathbb R\quad\longrightarrow\quad \text{BIM}^\times(M) $$ that, in the presence of a state (or weight), lifts to a homomorphism $i\mathbb R\to Aut(M)$. Here, $\text{BIM}^\times(M)$ denotes the 2-group of invertible $M$-$M$-bimodules. The bimodule $\mathbf{\Phi}(it)$ is the non-commutative $L^p$-space for the value $\frac 1 p=it$.

Given a ring $R$ of characteristic $p$, itsFrobeniusis a canonically defined homomorphism $$ \mathbf{F}:\mathbb N\quad\longrightarrow\quad End(R) $$ such that $\mathbf{F}(1)$ sends $x$ to $x^p$. More generally, $\mathbf{F}(n)$ sends $x$ to $x^{p^n}$.

So far, the only analogy I can see is that both $\mathbf{F}$ and $\mathbf{\Phi}$ are canonically defined actions...

In what sense does the Frobenius provide a spectral realization of a zeta-function?The eigenvalues of the Frobenius operator (acting on cohomology groups if you like) are the zeros of the $\zeta$-function, is this sufficient? $\endgroup$9more comments