2010/12/15 by Oliver Lorscheid, Lorscheid, Oliver
Mathematics · #11F67 #11M26 #11M32 #11M36 #11M38 #11R58 #20C08 #32N10 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F67 #msc:11M26 #msc:11M32 #msc:11M36 #msc:11M38 #msc:11R58 #msc:20C08 #msc:32N10
paper · pdf · doi:10.48550/arxiv.1012.3223
32 pages
arxiv created 2010/12/15 · arxiv updated 2010/12/16
The space of toroidal automorphic forms was introduced by Zagier in 1979. Let F be a global field. An automorphic form on \GL(2) is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight s is toroidal if s is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The (n-1)-th derivative of a non-trivial Eisenstein series of weight s and Hecke character χ is toroidal if and only if L(χ,s+1/2) vanishes in s to order at least n (for the "only if"-part we assume that the characteristic of F is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals h(g-1)+1 if the characterisitc is not 2; in characteristic 2, the dimension is bounded from below by this number. Here g is the genus and h is the class number of F. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.