Jacquet–Langlands correspondence

Jacquet–Langlands correspondence
Name	Jacquet–Langlands correspondence
Fields	Number theory; Representation theory; Automorphic forms

Jacquet–Langlands correspondence is a major theorem in the theory of automorphic forms and representation theory relating automorphic representations of GL(2) and of quaternion algebra groups. It sits at the intersection of the Langlands program, the theory of modular forms, and the trace formula, and connects work of Hervé Jacquet, Robert Langlands, and contemporaries with earlier developments by André Weil, Erich Hecke, and Atle Selberg. The correspondence has both local and global formulations and has influenced research around Shimura varieties, Galois representations, and the Arthur–Selberg trace formula.

Overview and Historical Background

The correspondence was formulated in the wake of classical results of Peter L. Taylor, Jean-Pierre Serre, and Goro Shimura on modular forms, as well as Hecke’s and Maass’s studies of zeta functions and non-holomorphic forms. It emerged from a collaboration between Hervé Jacquet and Robert Langlands during the 1970s that built on work of Carl Ludwig Siegel, Harish-Chandra, and André Weil concerning harmonic analysis on adelic groups and the distribution of automorphic spectra. Influences include the modularity investigations of Shimura and Taniyama, the adelic framework introduced by John Tate, and the representation-theoretic foundations developed by Roger Godement and Ilya Piatetski-Shapiro. Subsequent refinements and applications involved James Arthur, Benedict Gross, Michael Harris, Richard Taylor, and Robert Kottwitz.

Statement of the Correspondence

Roughly, the correspondence gives a bijection between irreducible automorphic representations of GL(2) over a number field and irreducible automorphic representations of the multiplicative group of a quaternion algebra that are discrete series at places where the algebra is ramified. The global statement uses adèles as in John Tate’s thesis and the automorphic representations studied by Jacquet, Piatetski-Shapiro, and Shalika; it relies on local correspondences at places studied by Harish-Chandra and Colin Bushnell. The matching preserves L-functions and epsilon factors in the sense of Godement–Jacquet and Deligne, and is compatible with the local Langlands correspondence proved by Gérard Laumon, Michael Harris, and Guy Henniart in later work.

Local and Global Aspects

Locally, the correspondence compares smooth representations of GL(2, F_v) with representations of D_v^× for a quaternion algebra D over a local field F_v as developed by Bushnell and Henniart; this uses the structure theory of reductive p-adic groups due to Jean-Pierre Serre and George Lusztig. Globally, the matching of automorphic spectra is organized via the adelic language of André Weil and the trace formula apparatus of James Arthur and Selberg. Ramification behavior follows patterns familiar from the work of John Milnor, Pierre Deligne, and Ken Ribet on conductors and local constants, while global multiplicity statements echo ideas from Robert Langlands’s original conjectures and contributions by Benedict Gross and Don Zagier.

Representation-Theoretic Framework

The correspondence is framed in the representation theory of reductive groups over local and global fields studied by Harish-Chandra, Roger Howe, and Alexandre Kirillov. Objects include cuspidal automorphic representations, discrete series, principal series, and newforms in the sense of Atkin and Lehner and Serre. Intertwining operators, parabolic induction, and characters play roles analogous to constructions in the works of I. M. Gelfand, David Kazhdan, and Masatoshi Kashiwara. Compatibility with Hecke algebras as studied by Keiichi Shimizu and Richard Taylor, and with epsilon factors from Tate and Deligne, is central to the representation-theoretic formulation.

Methods of Proof and Key Techniques

Jacquet and Langlands used the adelic trace formula and a careful study of spectral decompositions, building on Selberg’s trace formula and Arthur’s later stabilization techniques. Their arguments employ harmonic analysis on adele groups in the spirit of Tate’s thesis, orbital integrals in the sense of Jean-Loup Waldspurger and Robert Kottwitz, and comparison of local characters as developed by Harish-Chandra and Roger Howe. Later approaches and refinements have used p-adic methods influenced by Pierre Colmez, cohomological methods inspired by Michael Harris and Richard Taylor, and the stable trace formula ideas of James Arthur and Jean-Loup Waldspurger.

Examples and Explicit Constructions

Concrete instances include the classical correspondence between holomorphic newforms on congruence subgroups of SL(2, Z) studied by Atkin and Lehner and theta lifts related to Weil representations investigated by André Weil and Rolf Berndt. Quaternionic modular forms on Shimura curves of Goro Shimura and Yutaka Taniyama exhibit the correspondence to classical modular forms explored by Don Zagier and Benedict Gross. Explicit matching of Fourier coefficients and Hecke eigenvalues draws on computations by Fred Diamond, Jerry Shurman, and Ken Ono, while examples over totally real fields relate to work of Bas Edixhoven and Fred Diamond on Hilbert modular forms.

Applications and Consequences

The correspondence has consequences for the arithmetic of elliptic curves as in the modularity results pioneered by Andrew Wiles and Richard Taylor, for the construction of Galois representations following Deligne and Pierre Colmez, and for the study of special values of L-functions as in the Gross–Zagier formula. It informs the theory of Shimura varieties developed by Goro Shimura and Michael Harris, plays a role in the trace formula comparisons used by James Arthur in the stabilization of the trace formula, and connects to reciprocity principles envisioned by Robert Langlands and Pierre Deligne. Modern applications involve the work of Laurent Clozel, George Pappas, and Mark Kisin on arithmetic geometry and the p-adic Langlands program spearheaded by Matthew Emerton and Christophe Breuil.

Category:Number theory