I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
- Generates a copy of the integers (or rather their profinite completion) in local Galois groups. Geometrically, this gives us an "infinitesimal circle" above every point, for which Frobenius is the monodromy. E.g. an algebraic curve over a finite field can be thought of as a 3-manifold fibering over the circle, with Frobenius as the circle direction.
- Provides natural grading on "mixed" objects (e.g. cohomologies of singular schemes, perverse sheaves etc.) by its eigenvalues. Purity: only one graded piece. Key for results such as the Decomposition Theorem.
- Pullback of deRham complex becomes $\mathcal{O}$-linear. (Derivative of pth power is zero – simplifies calculus!) Leads to the Cartier operator on cohomology. Linear structure enables e.g. Deligne-Illusie to prove degeneration of Hodge-to-deRham.
- On same note, sheaves acquire a canonical connection after Frobenius pullback – when Frobenius is an isomorphism. Counterexample: $X$ abelian variety, $H^0(X,\Omega)$ in $H^1(X)$ (first piece of Hodge filtration) is killed by Frobenius – since every one-form is locally exact.
- Its failure to commute with a connection gives rise to p-curvature, a canonical $\mathcal{O}^p$ – linear form. This measures the obstruction to exponentiating a connection through the p-th order (p!=0 => need for divided powers..)
- p-curvature of the Gauss-Manin connection as Kodaira-Spencer class (Katz).
- Provides an automorphism of the additive group – generates ring of additive polynomials. Analog of ring of differential operators. Embeddings of coordinate rings into this ring (Drinfeld modules) provide important moduli problems, key to Langlands program for function fields.
- Gives rise to automorphisms of flag varieties (for example) => ask for a flag and its Frobenius translates to have prescribed relative position: Deligne-Lusztig varieties (Borel-independent analogs of Schubert cells).
- The kernels of Frobenius and its iterates give natural infinitesimal neighborhoods of the identity on group schemes.
- F-crystals: both "flat connection" and Frobenius structure. F is horizontal, but nonetheless eigenvalues not constant -> Newton polygons, and related stratifications of moduli schemes.
- As a contracting map – can apply fixed point formulas after applying Frobenius.
- Dwork's principle – use Frobenius to fix constant terms to do p-adic integration (use to patch together local pieces).
- Frobenius liftings to characteristic zero – p-adic analogs of Kahler metrics (Mochizuki).
- Together with Verschibung and homothety generates the ring of Cartier operators. Appropriate modules over this (Dieudonne modules) are essential to classification of group schemes.
I think it would be very interesting if some MO users (among whom maybe even the author himself) would
- Expand the description of each point 1.-10. also explaining his/her own intuitive understanding of the phenomenon, and giving some examples and reference to interested readers.
- Add new items to the list.
Best Answer
Pullback by the Frobenius realizes the p^th power Adams operation as an actual functor on the category of vector bundles over a scheme in characteristic p, something not possible in characteristic zero. This observation of Quillen was the starting point for his first approach to the (complex) Adams conjecture, details completed by Friedlander.