An automorphic form cusp in the classical theory of modular forms for a congruence subgroup of is a group orbit of on the projective line . Adjoining a point for each such group orbit to the quotient of the upper half-plane by gives the compact modular curve. For the full modular group Gamma, all elements of are equivalent to , so there is only one cusp (Diamond and Shurman 2005).

The expansion at a cusp is obtained by moving a representative to and expanding the transformed form in a local Fourier series. A cusp form has zero constant coefficient in this expansion at every cusp (Diamond and Shurman 2005).

In the corresponding three-dimensional setting, let be an imaginary quadratic field with ring of integers . For a finite-index subgroup of acting on three-dimensional hyperbolic space, the cusps of the quotient are the group orbits of on the projective line (Elstrodt et al. 1998). This is the setting for the cusp expansions of Kubota's cubic theta function over (Dunn and Radziwiłł 2024).

See also

Cusp Form, Modular Curve, Modular Form

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References

Diamond, F. and Shurman, J. A First Course in Modular Forms. New York: Springer, 2005.Dunn, A. and Radziwiłł, M. "Bias in Cubic Gauss Sums: Patterson's Conjecture." Ann. Math. 200, 967-1057, 2024. https://doi.org/10.4007/annals.2024.200.3.3.Elstrodt, J.; Grunewald, F.; and Mennicke, J. Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory. Berlin, Germany: Springer, 1998. https://doi.org/10.1007/978-3-662-03626-6.

Cite this as:

Weisstein, Eric W. "Automorphic Form Cusp." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AutomorphicFormCusp.html

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