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| Maass cusp form | |
|---|---|
| Name | Maass cusp form |
| Field | Number theory, Representation theory, Mathematical analysis |
| Introduced by | Hans Maass |
| Introduced year | 1949 |
Maass cusp form.
Maass cusp forms are nonholomorphic automorphic eigenfunctions of the hyperbolic Laplacian on the upper half-plane that vanish at cusps and play a central role in analytic number theory, spectral theory, and representation theory. They connect the work of Hans Maass with the theory of Atkin–Lehner, Hecke operators, and the Selberg trace formula, and they feature in conjectures of Sato–Tate, Langlands, and Quantum unique ergodicity.
A Maass cusp form is a smooth, square-integrable function on the quotient of the upper half-plane by a discrete subgroup such as SL(2,Z), invariant under the action of that subgroup, annihilated by the hyperbolic Laplacian, and rapidly decaying at the cusps. For congruence subgroups like Gamma0(N), Maass cusp forms can be chosen to be simultaneous eigenfunctions of the full commutative algebra generated by Hecke operators and by the Laplacian, giving rise to arithmetic eigenpackets tied to automorphic representations for GL(2). Eigenvalues lie in the continuous and discrete spectrum described by the Selberg eigenvalue conjecture and are related to spectral parameters used in harmonic analysis on Riemann surfaces and locally symmetric spaces.
Explicit Maass cusp forms are rare; classical examples include those constructed via Poincaré series and by lifting procedures such as the theta correspondence and Maaß–Selberg. Constructed families appear for congruence groups like Gamma1(N), Gamma0(N), and for Bianchi groups over imaginary quadratic fields studied by Hecke and Maaß. Other constructions use Eisenstein series regularization from Atkin–Lehner theory or via trace formulas from Selberg trace formula, and specialized examples arise in the work of Hecke on Hecke characters and in the computational tables of Andrew Booker and A. Booker and A. Strömbergsson.
Spectral decomposition on spaces of Maass cusp forms uses the Selberg trace formula, connecting discrete eigenvalues to lengths of closed geodesics on modular curves and to scattering matrices from Eisenstein series. The discrete spectrum contains cusp forms and possibly residual spectra tied to Eisenstein series poles studied by Harish-Chandra and Langlands. Important conjectures involve the size of the lowest nonzero eigenvalue (the Selberg eigenvalue conjecture) and quantum limits as in Quantum unique ergodicity proved in special cases by Lindenstrauss and Soundararajan. Relations to representation theoretic notions appear via the classification of unitary representations of SL(2,R) and the correspondence with principal series and discrete series representations developed by Gelbart and Jacquet.
At a cusp, a Maass cusp form admits a Fourier expansion with coefficients indexed by integers, involving K-Bessel functions parameterized by the spectral parameter. The Fourier coefficients are linked to Hecke eigenvalues when the form is an eigenfunction of Hecke operators; these coefficients satisfy multiplicative relations analogous to those for modular forms proven in the theory of Hecke algebras. Growth estimates and bounds such as subconvexity bounds for associated L-functions rely on uniform control of K-Bessel decay, techniques from Iwaniec and Sarnak, and mean value estimates from the theory of automorphic forms on GL(2). Ramanujan–Petersson type bounds for Maass forms are conjectural in the general nonholomorphic setting and relate to the Ramanujan conjecture within the Langlands program.
To a Hecke–eigen Maass cusp form one attaches an L-function via a Dirichlet series formed from Fourier–Hecke coefficients; these L-functions satisfy analytic continuation and a functional equation deriving from the action of the Fricke involution and from global functional equations in the Langlands program. Rankin–Selberg convolution with holomorphic forms or other Maass forms produces L-functions studied by Rankin, Selberg, and Jacquet–Langlands, with central value conjectures connected to the Birch and Swinnerton-Dyer conjecture analogues and to subconvexity problems addressed by Michel and Venkatesh. Nonvanishing and special value results use periods, Waldspurger formulae, and the relative trace formula developed by Kuznetsov and Jacquet.
Maass cusp forms influence arithmetic via trace formula applications to counting problems, equidistribution results on modular curves, and relations to class numbers and representations by quadratic forms as in the work of Siegel and Duke. They appear in the spectral side of pretrace formulas used to study arithmetic quantum chaos, linking to conjectures by Rudnick and Sarnak. Connections with automorphic representations for GL(2) enable transfer results, functorial lifts, and relations to symmetric power L-functions appearing in Gelbart–Jacquet and Kim–Shahidi works. Arithmeticity results for lattices like Margulis's theorem interplay with the spectral theory of Maass forms on arithmetic locally symmetric spaces.
Numerical computation of Maass cusp forms employs methods such as the Hejhal algorithm, Kuznetsov trace formula based approaches, and finite element or spectral collocation techniques implemented by researchers like Hejhal, Strombergsson, Booker, and Venkatesh. Large-scale computations for SL(2,Z) and congruence subgroups have produced eigenvalue tables and Fourier coefficient data used to test conjectures like Ramanujan and quantum ergodicity. Known rigorous results include Weyl laws for eigenvalue counting via Selberg trace formula and proven instances of quantum unique ergodicity for arithmetic surfaces by Lindenstrauss and hybrid bounds for L-functions by Blomer and Harcos.