From quadratic polynomials and continued fractions to modular forms

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 prove two different such representations, which imply the exponential convergence of the sums.

For Logic Seminar: Note change of time and location!