naa a22 4500 51078349X CHVBK 20180411083254.0 cr unu---uuuuu 180411e20130301xx s 000 0 eng 10.1007/s11856-012-0084-2 doi (NATIONALLICENCE)springer-10.1007/s11856-012-0084-2 Lorscheid Oliver Math. Dept., The City College of New York, 160 Convent Ave., 10031, New York, NY, USA aut Toroidal automorphic forms for function fields [Elektronische Daten] [Oliver Lorscheid] 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 x is toroidal if and only if L(x, 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 characteristic 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. Hebrew University Magnes Press, 2012 Israel Journal of Mathematics Springer US; http://www.springer-ny.com 194/2(2013-03-01), 555-596 0021-2172 194:2<555 2013 194 11856 https://doi.org/10.1007/s11856-012-0084-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11856-012-0084-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Lorscheid Oliver Math. Dept., The City College of New York, 160 Convent Ave., 10031, New York, NY, USA aut NATIONALLICENCE 773 0- Israel Journal of Mathematics Springer US; http://www.springer-ny.com 194/2(2013-03-01), 555-596 0021-2172 194:2<555 2013 194 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer