University of Illinois at Urbana-Champaign
Department of Mathematics
Colloquium
by
Prof. Alexandru Buium

4:00 PM, Thursday, September 11, 1997, 245 Altgeld Hall.

An Arithmetic Analogue of Differential Equations
An arithmetic analogue of derivations on number fields is introduced and a theory of (non-linear) algebraic differential equations is developped. The key concept is that of "arithmetic jet space" and the main purpose is the study of its geometry. Applications include an effective version of the Manin-Mumford conjecture an arithmetic analogue of Manin's theorem of the kernel and a theory of modular forms with "differential weights". 

Background References:


(1) Lang, Number Theory III, Encyclopaedia of Math. Sciences, vol. 60,
    Springe 1991. This is also just published in paperback by Springer as
    "Survey of Diophantine Geometry". In chapter 6 the "Manin kernel" is
    mentioned.


(2) Lang, Introduction to modular forms, Springer 1995.