Toroidal automorphic forms, Waldspurger periods and double Dirichlet series

The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL 2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a nontrivial zero of the zeta function of the quadratic field corresponding to the torus. In this chapter, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that this space admits a decomposition into a subspace of Eisenstein series (and derivatives) and a subspace of cusp forms. The subspace of Eisenstein series is generated by all derivatives up to order n-1 of an Eisenstein series of weight s and class group character ω for certain n, s, ω, namely, precisely when s is a zero of order n of the L-series L F (ω, s). The subspace of cusp forms consists of exactly those cusp forms π whose central L-value is zero: 퐿(휋,1/2)=0. The proofs are based on an identity of Hecke for toroidal integrals of Eisenstein series and a result of Waldspurger about toroidal integrals of cusp forms combined with nonvanishing results for twists of L-series proven by the method of double Dirichlet series.