Premium
A Classical Characterization of Newforms with Equivalent Eigenforms in S k + 1/2 (4N, χ)
Author(s) -
Frechette Sharon M.
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703004666
Subject(s) - mathematics , cusp (singularity) , modulo , pure mathematics , cusp form , dirichlet distribution , quadratic equation , characterization (materials science) , character (mathematics) , modular form , space (punctuation) , combinatorics , mathematical analysis , geometry , linguistics , philosophy , materials science , boundary value problem , nanotechnology
The paper continues the investigation of the Hecke structure of spaces of half‐integral weight cusp forms S κ + 1/2 (4 N , χ), where k and N are positive integers with N odd, and χ is an even quadratic Dirichlet character modulo 4 N . In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non‐twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in S κ + 1/2} (4 N , χ). The result reformulates, in classical terms, the representation‐theoretic conditions given by Flicker and Waldspurger. The conditions involve easily‐verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half‐integral weight cusp forms in any space S κ + 1/2 (4 N , χ) as above.