 
Summary: Modular Equations and
Special Function Transformations
Rob Maier
Depts. of Mathematics and Physics
University of Arizona
Geometry, Topology and Physics Seminar
UCSB
8 February, 2008
HighLevel Overview
· PicardFuchs equations:
Let V
X x be a family of ndimensional algebraic varieties,
with smooth fibres over the complement of a finite set.
Choose a holomorphic nform on a smooth fibre V0 V, and
ncycles 1, . . . , r that give a basis for its nth homology.
Then can be extended to a meromorphic family of nforms (x),
and the cycles (homology classes) to (multivalued!) functions of x.
The periods i(x)
(x) are multivalued too, but satisfy a Fuchsian
