Milner 317

Wednesdays, 12:30-1:30 PM

Texas A&M University

**Wednesday, October 18
**

**Title: ** Linear
relations and congruences for the coefficients of cusp forms

**Abstract:** Modular forms
are holomorphic functions on the upper half plane that transform nicely
under certain subgroups of $SL_2(Z)$. One consequence of that is that
they have an expansion in powers of $e^{2\pi i z}$. The study of the
arithmetic properties of the coefficients of this expansion is a very
rich branch in the theory of modular forms. In this talk, we discuss
the question of divisibility of the coefficients by a prime $p$. In
particular, we show criteria for $p$ being a non-ordinary prime for a
Hecke eigenform $f$ (which means that $p$ divides the coefficient of
$e^{2 \pi i p z}$ in the expansion of $f$. We'll also discuss linear
relations satisfied by the coefficients of modular forms.