CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS
Abstract
It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.
References
- Math. Ann. 185, 134 (1970), DOI: 10.1007/BF01359701. Crossref, ISI, Google Scholar
- Y. J. Choie and W. Kohnen, Diagonalizing "bad" Hecke operators on spaces of cusp forms, in Number Theory, Dev. Math., Vol. 15 (Springer, New York, 2006), pp. 23–26; MR MR2213826 (2006m:11056) . Google Scholar
- Math. Ann. 212, 285 (1975), DOI: 10.1007/BF01344466. Crossref, ISI, Google Scholar
- J. Number Theory 112(2), 298 (2005), DOI: 10.1016/j.jnt.2004.10.009. Crossref, ISI, Google Scholar
- Trans. Amer. Math. Soc. 331(1), 425 (1992). Crossref, ISI, Google Scholar
- G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45(3) (1978) 637–679; MR507462 (80a:10043); Errata, ibid., 48(3) (1981) 697; MR630592 (82j:10051) . Google Scholar
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