The Ramanujan Journal

, Volume 30, Issue 2, pp 257–278 | Cite as

Newform theory for Hilbert Eisenstein series



In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than \(\frac{1}{2}\). Additionally, we provide a number of applications of this newform theory. Let Open image in new window denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level \(\mathcal{N}\) and Hecke character Ψ over a totally real field K. For any prime \(\mathfrak{q}\) dividing \(\mathcal{N}\), we define an operator \(C_{\mathfrak{q}}\) generalizing the Hecke operator \(T_{\mathfrak{q}}\) and prove a multiplicity-one theorem for Open image in new window with respect to the algebra generated by the Hecke operators \(T_{\mathfrak{p}}\) ( \(\mathfrak{p}\nmid\mathcal{N}\)) and the operators \(C_{\mathfrak{q}}\) (\(\mathfrak{q}\mid\mathcal{N}\)). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).


Hilbert modular form Eisenstein series Newform 

Mathematics Subject Classification (2000)

11F41 11F11 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Parametric Portfolio AssociatesSeattleUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA

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