cuspformar

Cuspformar is a term used in some discussions of modular form theory to denote a family of cusp forms augmented by an auxiliary deformation parameter, ar. It is not a standard object in mainstream texts, but it appears in certain exploratory or expository writings as a way to describe deformations or interpolations of cusp forms and their L-functions.

Definition: For a fixed weight k and a congruence subgroup Gamma of SL2(Z), a cuspformar f is a cusp form in the usual sense: holomorphic on the upper half-plane, transforming by f((a z+b)/(c z+d)) = (cz+d)^k f(z) for all gamma in Gamma, and vanishing at all cusps. The ar-parameter is introduced by allowing the Fourier coefficients a_n to depend analytically on ar, so f(z) = sum_{n≥1} a_n(ar) e^{2π i n z}, with a_n(ar) analytic in ar near 0. The associated L-function L(s, f, ar) = sum a_n(ar) n^{-s} is considered for values of ar where the Dirichlet series converges and can often be continued analytically, sometimes satisfying a functional equation that varies with ar.

Construction and examples: A typical construction is ar-twisting, where a_n(ar) = a_n n^{-ar} for a fixed cusp form f, or regularization procedures that produce ar-dependent coefficients. For ar=0, one recovers the original cusp form. Such deformations are discussed as heuristic tools for studying families of L-values and p-adic interpolations.

Notes: The use of cuspformar is informal in many sources, and it is not universally adopted as a standard object.