Breadcrumb
From quadratic polynomials and continued fractions to modular forms
Wed 22 May 2013, 16:00
Paloma Bengoechea
College de France, Paris
Organisers: Tim Dokchitser, Daniel Loughran
ABSTRACT
Zagier studied in 1999 certain real functions defined in a very simple way
as sums of powers of quadratic polynomials with integer coefficients.
These functions give the even parts of the period polynomials of the
modular forms which are the coefficients in Fourier expansion of the
kernel function for Shimura-Shintani correspondence. He conjectured for
these sums a representation in terms of a finite set of polynomials coming
from reduction of binary quadratic forms and the infinite set of
transformations occuring in a continued fraction algorithm of the real
variable. We will give two different such representations, which imply the
exponential convergence of the sums. The expressions we prove arise from
more general results on polynomials of arbitrary even degree.