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| Jacobi form | |
|---|---|
| Name | Jacobi form |
| Field | Mathematics |
Jacobi form is a class of automorphic objects arising in the intersection of modular form theory, elliptic function theory, and the representation theory of Heisenberg group extensions of modular group. Jacobi forms generalize classical theta functions and link the theory of elliptic curves, Siegel modular forms, and various arithmetic and geometric structures appearing in the work of Jacques Hadamard-era analysts and modern researchers such as Martin Eichler, Don Zagier, and Igor Frenkel. They appear naturally in problems involving quadratic forms, lattices, and the study of automorphic representations for adelic groups like SL2 and symplectic groups.
A Jacobi form on the complex upper half-plane with elliptic variable is a holomorphic function satisfying specific transformation rules under the action of SL2(Z) and a semidirect product with a discrete Heisenberg group lattice; it is typically specified by a weight and an index, and admits a Fourier expansion reflecting periodicity in the elliptic variable. Key properties include modularity under congruence subgroups such as Γ0(N), holomorphicity conditions at cusps related to growth conditions analogous to those for classical cusp forms, and a close relationship with theta series attached to positive definite quadratic forms or even unimodular lattices like the E8 lattice or Leech lattice.
The theta decomposition expresses a Jacobi form as a finite linear combination of vector-valued theta functions times modular forms on SL2(Z), linking coefficients to representation numbers of binary quadratic forms and lattice counts. Its Fourier expansion in variables corresponding to the modular curve coordinate and the elliptic coordinate yields Fourier–Jacobi coefficients that are modular forms for stabilizer subgroups such as Sp2(Z) components and encode arithmetic data like Fourier coefficients appearing in generating functions studied by Ramanujan, Poincaré, and Hecke.
Jacobi forms transform with a factor of automorphy under the action of SL2(Z) and additional translations coming from a discrete Heisenberg group related to the integral lattice Z^2; these transformation laws involve the weight and index and are compatible with the metaplectic double cover appearing in the theory of Weil representations. The transformation behavior under elements of congruence subgroups such as Γ1(N) and their lift to Mp2(R) plays an essential role in constructing theta lifts to groups like Sp4(Z) and in the correspondence between Jacobi forms and vector-valued modular forms described in the work of Gritsenko and Borcherds.
Classical examples include the Jacobi theta function and index-one Jacobi forms yielding elliptic genera such as the Witten genus and elliptic genus of K3 surfaces, while higher-index examples arise from theta series of positive-definite even lattices like E8 and the Niemeier lattices. Special cases cover holomorphic Jacobi forms, weak Jacobi forms, and mock Jacobi forms introduced in relation to the Moonshine phenomena studied by Conway and Norton and later by Eguchi and Hohenegger. The notion of cusp Jacobi forms parallels Siegel cusp forms and is exemplified by forms constructed by Skoruppa and Zagier.
Jacobi forms provide generating functions for representation numbers of quadratic forms, partition-theoretic functions studied by Hardy and Ramanujan, and traces of singular moduli central to the work of Gross and Zagier. They serve as kernels in theta lift constructions connecting automorphic forms on SL2 to automorphic representations of symplectic groups such as Sp4 and higher Siegel modular groups, underpinning results by Howe and Piatetski-Shapiro. In representation theory, Jacobi forms realize modules for the Heisenberg algebra and afford explicit models for the Weil representation and for spaces of vector-valued modular forms relevant to the theory of vertex operator algebras as in the work of Frenkel, Lepowsky, and Meurman.
Fourier–Jacobi expansions identify Siegel modular forms on Sp2n(Z) with sequences of Jacobi forms, enabling dimension formulas and lifting constructions such as the Maass lift and the Saito–Kurokawa lift studied by Maass and Kurokawa. Mock Jacobi forms arise in the context of mock modular forms investigated by Zwegers and Bringmann, linking nonholomorphic completions to harmonic Maass forms and to quantum invariants in topology studied by Lawrence and Zagier. These connections have been exploited in moonshine-type correspondences involving groups like the Monster group and sporadic groups including Mathieu group M24.