Concept

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group acts on a complex-analytic manifold . Then, also acts on the space of holomorphic functions from to the complex numbers. A function is termed an automorphic form if the following holds: where is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of . The factor of automorphy for the automorphic form is the function . An automorphic function is an automorphic form for which is the identity. Some facts about factors of automorphy: Every factor of automorphy is a cocycle for the action of on the multiplicative group of everywhere nonzero holomorphic functions. The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form. For a given factor of automorphy, the space of automorphic forms is a vector space. The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding factors of automorphy. Relation between factors of automorphy and other notions: Let be a lattice in a Lie group . Then, a factor of automorphy for corresponds to a line bundle on the quotient group . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle. The specific case of a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (1)
MATH-603: Subconvexity, Periods and Equidistribution
This course is a modern exposition of "Duke's Theorems" which describe the distribution of representations of large integers by a fixed ternary quadratic form. It will be the occasion to introduce the
Related lectures (1)
Fourier Expansion and Quaternion Algebra
Discusses Fourier expansion, quaternion algebra, and Whittaker model in automorphic forms.
Related publications (3)

Algebraic twists of GL(2) automorphic forms

Vignesh Arumugam Nadarajan

In this thesis we consider the problem of estimating the correlation of Hecke eigenvalues of GL2 automorphic forms with a class of functions of algebraic origin defined over finite fields called trace functions. The class of trace functions is vast and inc ...
EPFL2023

Sign changes of Kloosterman sums with almost prime moduli

Ping Xi

We prove that the Kloosterman sum changes sign infinitely often as runs over squarefree moduli with at most 10 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler and Matomaki, replacing 10 by 23, 18 and 15, respectively ...
Springer Verlag2015

Hybrid Bounds For Automorphic Forms On Ellipsoids Over Number Fields

Philippe Michel

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds X of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realiz ...
Cambridge Univ Press2013
Related concepts (2)
Felix Klein
Christian Felix Klein (klaɪn; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents.
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.