Toroidal Automorphic Forms for Some Function Fields
نویسنده
چکیده
Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g ≤ 1, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an “automorphic” proof for the Riemann hypothesis for the zeta function of those curves.
منابع مشابه
Toroidal Automorphic Forms, Waldspurger Periods and Double Dirichlet Series
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL2-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 non-trivial zero of the zeta function of the q...
متن کاملConstruction of Vector Valued Modular Forms from Jacobi Forms
We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.
متن کاملComputing Modular Forms for GL2 over Certain Number Fields
The cohomology of an arithmetic group is built out of certain automorphic forms. This allows computational investigation of these automorphic forms using topological techniques. We discuss recent techniques developed for the explicit computation of the cohomology of congruence subgroups of GL2 over CM-quartic and complex cubic number fields as Hecke-modules.
متن کاملA brief overview of modular and automorphic forms
These notes were originally written in Fall 2010 to provide a very quick overview of some basic topics in modular forms, automorphic forms and automorphic representations. I have not made any significant changes since, or even proofread them completely (so some information may be outdated, and errors may remain), mostly just corrected some typos. If you spy any more errors, or have suggestions,...
متن کاملGauge Couplings and their M – Theory Origin
We work out the relation between automorphic forms on SO(2+s, 2, Z) and gauge one– loop corrections of heterotic K3×T 2 string compactifications for the cases s = 0, 1. We find that one–loop gauge corrections of any orbifold limit of K3 with arbitrary choice of gauge bundles can always be expressed by their instanton numbers and generic automorphic forms. These functions classify also one–loop ...
متن کامل