### School of Mathematics and Statistics

### Mark Kisin

University of Sydney

### Unit *F*-Crystals and a Conjecture of Katz.

**Friday 15th October, 12-1pm, Carslaw 375. **
Let *X* be a variety over a finite field *k* of
characteristic *p*. One can attach to X its "arithmetic
fundamental group" *G(X).*

Let *l* be a prime. If *R: G(X) --->
GL(Z*_{l}) is a representation of
*G(X),* then one can attach to *R* two
power series with *Z*_{l} coefficients: Its usual
*L*-function *L(X,R),* which carries information
about the arithmetic of *R* and *X,* and a
cohomological *L*-function *L(k, f*_{!}R)
which has the advantage that it is a rational function.

For *l* different from *p* a theorem of Grothendieck
says that these two are equal. When *l=p* a conjecture of
Katz predicts that the quotient
*L(X,R)/L(k,f*_{!}R) is an invertible analytic
function on the *p*-adic unit disk.

After explaining the *L*-functions mentioned above, I will
report on the recent proof of Katz's conjecture by M. Emerton and
myself.