Algebra Seminar

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.