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Daquing Wan,
University of California, Irvine
``Introduction To Dwork's Conjecture''

February 18, 1999

In this lecture, I shall give an elementary and self-contained
introduction to Dwork's conjecture (1973) on the p-adic meromorphic
continuation of his unit root L-function L(f,T) attached to a family
f: Y->X of algebraic varieties over a finite field of characteristic p.
This conjecture is a p-adic extension of the Weil conjecture (1949) from
a single variety (or a family of zero-dimensional varieties) to an
arbitrary family of varieties.
When f is a family of **zero**-dimensional
varieties (i.e., f is a finite map),
the unit root L-function L(f, T) becomes the zeta function Z(Y,T) of the
variety Y. The zeta function Z(Y,T) is a generating function which counts
the number of rational points on Y. The zeta function Z(Y,T) is a rational
function as conjectured by Weil and first proved by Dwork (1960) using
p-adic analytic method. The key step in Dwork's proof is to show that
Z(Y,T) is p-adic meromorphic. The zeros and poles of the zeta function
satisfy a suitable Riemann hypothesis as also conjectured by Weil but
proved by Deligne (1974-1980) using etale cohomology.

When f is a family of **positive**-dimensional varieties, the unit root
L-function L(f, T) is no-longer a rational function and the situation
is much more mysterious. But Dwork conjectured that the L-function is
p-adic meromorphic. The simplest example is the universal family f_E
of elliptic curves. In this case, the L-function L(f_E,T) is known to
be p-adic meromorphic (Dwork, 1971). However, even in the elliptic
family case, very little is known about the absolute values of the
zeros of L(f_E,T), namely, the p-adic Riemann hypothesis for L(f_E,T),
which contains important arithmetic information about modular forms
such as the Gouvea-Mazur conjecture and the p-adic Ramanujuan-Peterson
conjecture.

A note from the colloquium chair: I highly recommend the
Dwork memorial
article which was written for the March, 1999
*Notices* by Nick Katz
and John Tate.