A peculiar modular form of weight one, by Stephen S. Kudla, Michael Rapoport, and Tonghai Yang
In this paper we construct a modular form f of weight one
attached to an imaginary quadratic field K. This form,
which is non-holomorphic and not a cusp form, has several curious
properties. Its negative Fourier coefficients are non-zero precisely
for neqative integers -n such that n >0 is a norm from K, and these
coefficients involve the exponential integral. The Mellin transform
of f has a simple expression in terms of the Dedekind
zeta function of K and the difference of the logarithmic derivatives
of Riemann zeta function and of the Dirichlet L-series of K.
Finally, the positive Fourier coefficients of f are connected with the
theory of complex multiplication and arise in the work of Gross and Zagier
on singular moduli.
Stephen S. Kudla, Michael Rapoport, and Tonghai Yang <email@example.com>