(This will be a talk about work in progress.)
Motivated from his p-adic study of the variation of the zeta function as the variety moves through a
family, Dwork conjectured that a new type of L-function, the so-called unit root L-function, was always
p-adic meromorphic. In the late 1990s, Wan proved this using the theory of sigma-modules, demonstrating
that unit root L-functions have structure. Little more is known.
The unit root L-function coming from the Legendre family of elliptic curves is connected with p-adic
modular forms. After briefly discussing this, we will consider a similar (conjectural) statement for the unit
root L-function coming from the Kloosterman family of exponential sums and p-adic Hilbert modular forms.