Alladi Ramakrishnan Hall

The geometry of the upper half-plane over the field with p elements

L. Merel

University of Paris VII

Let p be a prime number. B. Edixhoven has a discovered curious relation in the
representation theory of GL_2(F_p), which can be restated in the geometry of the
space formed by a field with p^2 elements deprived of its subfield with p elements
(the « upper half-plane» over F_p). A theorem of Chen on certain isogenies of modular
abelian varieties follows. We will not discuss Chen’s theorem until lecture 4 but we
will explain the signification of Edixhoven's relation for congruence subgroups of
SL_2(Z) and its application for modular curves and modular forms.

Prerequisite : Nothing but basic group theory for the first part. Some knowledge of
modular forms will be useful for the second part. However all concepts will be
recalled.