hecke

Hecke is a German surname that appears in several areas of mathematics. In mathematical literature, the name is most closely associated with the German number theorist Erich Hecke, whose work in modular forms and L-functions has had a lasting influence. Concepts bearing the Hecke name span number theory, representation theory, and geometry.

Hecke operators are a family of commuting linear operators acting on spaces of modular forms or automorphic forms. They enable the decomposition of these spaces into simultaneous eigenforms; the eigenvalues are related to Fourier coefficients and to Euler products in L-functions. The Hecke algebra is the algebra generated by all such operators and plays a central organizing role in modern number theory and the theory of automorphic representations.

Hecke characters, or Grössencharakters, generalize Dirichlet characters to number fields. They define Hecke L-functions, extending Dirichlet L-functions and providing important tools in class field theory and the Langlands program. These L-functions encode arithmetic information of number fields and are studied for their analytic properties and special values.

Hecke groups, also called Hecke triangle groups, are a family of Fuchsian groups generated by two Möbius transformations. They depend on a parameter and include the modular group as a special case. They arise in the study of hyperbolic geometry, modular forms, and spectral theory of automorphic forms.

Beyond these concepts, the surname Hecke appears in various mathematical contexts associated with different individuals, reflecting a broader influence in algebra, analysis, and geometry.