The Frobenius Page: A collection of one-liners and pithy sayings
about Frobenius, as a guide to the perplexed about this most
mysterious of operators (especially myself). Please
email me with comments,
corrections and especially suggestions for more items!
- 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 circel 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
- Pullback of deRham complex becomes 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
- Its failure to commute with a connection gives rise to p-curvature,
a canonical 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
- 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
- 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 (among other things!)