The Lam\'{e} differential equation $L_n(y)=0$ is the second order linear differential equation

Alexa van der Waall, University of Utrecht

The Lam\'{e} differential equation $L_n(y)=0$ is the second order
linear differential equation

$p(z)y''(z)+{1/2}p'(z)y'(z)-(n(n+1)+B)y(z)=0$

for a certain polynomial $p(z)$ of degree $3$ in $z$ and constants $n$
and $B$.  There exist cases in which the solution space of the Lam\'{e}
equation only consists of algebraic solutions.  A number of qualitative
results on this subject have been given by F. Baldassari and B.
Chiarellotto.  Some of their results will be mentioned in this talk.
The exploration of the notion of a monodromy group yields some explicit
algorithms for the construction of the Lame\'{e} equations with only
algebraic solutions.  We shall descibe at least one of these algorithms
and show how it works.