Abstract:
Modular forms are complex analytic objects that encode arithmetic
data coming from elliptic curves and other arithmetic geometric sources.
It is also possible to do arithmetic with the modular forms themselves and
this has important consequences for the arithmetic data they encode. By
looking at concrete examples, I will discuss the evolution of the concept
of congruences and padic variation of modular forms and will describe
some of the many consequences for the theory of Lfunctions. In
particular, I will explain briefly how padic Lfunctions vary
over the ColemanMazur eigencurve and describe recent numerical
experiments with Robert Pollack that produce new examples of padic
Lfunctions. The lecture will be aimed at graduate students and
nonspecialists except for a few remarks near the end.
