Hilbert modular forms

Hilbert modular forms
Name	Hilbert modular forms
Field	Number theory
Introduced	19th century
Major contributors	David Hilbert, Erich Hecke, Hermann Minkowski, Harold Stark, Goro Shimura, André Weil, John Tate, Friedrich Hirzebruch, Don Zagier, Robert Langlands, Gérard Laumon, Richard Taylor, Jean-Pierre Serre, Benedict Gross, Ken Ribet, Barry Mazur, Michael Harris

Hilbert modular forms are automorphic forms attached to totally real number fields that generalize classical Modular forms over Q. They connect the arithmetic of algebraic number fields, the representation theory of adele groups such as GL(2), and the geometry of Hilbert modular varieties like Hilbert modular surfaces and higher-dimensional analogs. These objects play central roles in modern approaches to the Langlands program, explicit class field theory, and the study of arithmetic of elliptic curves and abelian varietys with real multiplication.

Introduction

Hilbert modular forms arise from the action of arithmetic groups associated to a totally real field K such as Q(√5), Q(√2), Q(√13), or higher-degree fields studied by David Hilbert and Hermann Minkowski. They generalize classical constructions used by Erich Hecke and connect to automorphic representations considered by Gelfand, Graev, and Piatetski-Shapiro as well as later developments by Goro Shimura and André Weil. Their theory interfaces with objects like Hilbert class fields, CM fields, and Shimura varietys studied by Pierre Deligne and Jean-Pierre Serre.

Definition and Examples

For a totally real field K of degree n over Q, Hilbert modular forms are holomorphic functions on n copies of the upper half-plane transforming under the Hilbert modular group associated to K such as SL2 over the ring of integers OK or its congruence subgroups studied by Atkin, Lehner, Serre and Deligne-Serre. Typical examples include parallel-weight forms arising from theta series of positive definite quadratic forms considered by Carl Gustav Jacob Jacobi and Srinivasa Ramanujan-inspired constructions, base change forms from classical Modular forms via work of Langlands and Arthur, and forms attached to CM abelian varietys studied by Hecke and Weil. Explicit examples were computed by Hirzebruch and Zagier on Hilbert modular surfaces over fields like Q(√5).

Modular Groups and Hilbert Modular Surfaces

The Hilbert modular group for K, often GL2(OK) or SL2(OK), acts on a product of n copies of the upper half-plane giving quotient spaces that are Hilbert modular varieties. For n=2 these are Hilbert modular surfaces analyzed by Friedrich Hirzebruch, Don Zagier, Jarvis, and Freitag. Compactifications using techniques of Baily-Borel, Toroidal compactifications, and work of Mumford relate these quotients to algebraic surfaces, linking to the classification by Enriques and surfaces studied by Kodaira. The geometry of cusps and resolutions involves contributions from Eisenstein series studied by Hecke and compactification methods advanced by Ash, Mumford, Rapoport, and Tai.

Fourier Expansions and Hecke Operators

Hilbert modular forms admit Fourier expansions at cusps indexed by fractional ideals of OK analogous to q-expansions in classical theory used by Ramanujan and Hecke. The Fourier coefficients encode arithmetic invariants and are acted upon by Hecke operators constructed by double cosets in GL2(Adeles_K)) following the frameworks of Hecke, Atkin-Lehner, Jacquet-Langlands, and Iwaniec. Eigenforms yield automorphic representations studied by Gelbart, Jacquet, and Langlands, and multiplicity-one results employ methods of Godement and Jacquet-Shalika. Explicit computational work on Fourier coefficients and eigenvalues has been advanced by Dembélé, Greenberg, Buzzard, and Bringmann.

L-functions and Arithmetic Applications

L-functions attached to Hilbert modular eigenforms generalize classical L-series of Dirichlet and Hecke, satisfying analytic continuation and functional equations predicted by the Langlands conjectures and proven in special cases by Shimura, Jacquet, Shalika, Harris-Taylor, and Laumon. These L-functions relate to arithmetic questions about Elliptic curves and Abelian varietys with real multiplication, linking to modularity results of Wiles, Taylor, Breuil, Conrad, Diamond, and Kisin. Applications include explicit class field theory over totally real fields, constructions of p-adic L-functions by Perrin-Riou and Katz, and arithmetic of Hilbert Blumenthal abelian varietys studied by Deligne-Ribet.

Cohomological and Geometric Interpretations

Cohomological approaches view Hilbert modular forms as classes in the Betti, de Rham, or étale cohomology of Hilbert modular varieties, using techniques from Hodge theory and the theory of motives developed by Grothendieck, Deligne, and Scholl. The Eichler-Shimura correspondence and generalizations by Harder and Faltings relate forms to Galois representations constructed by Tate and Fontaine, with congruence phenomena studied by Mazur, Ribet, and Diamond. Geometric incarnations include moduli of abelian varieties with real multiplication and level structures investigated by Mumford, Rapoport, and Zink.

Historical Development and Key Results

The subject traces to foundational work of David Hilbert and Erich Hecke, with substantial 20th-century advances by Hermann Minkowski, Friedrich Hirzebruch, Goro Shimura, André Weil, and Harold Stark. Major milestones include the formulation of Hilbert modular surfaces by Hirzebruch and compactification results of Baily and Borel, automorphic representation frameworks by Langlands and Jacquet, modularity lifting techniques by Wiles and Taylor-Wiles, and p-adic developments by Hida and Katz. Contemporary progress ties to the work of Harris, Taylor, Laumon, Emerton, Kisin, Calegari, and Geraghty on modularity, Galois deformation, and reciprocity in the Langlands program.

Category:Number theory