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Newform theory for Hilbert Eisenstein series. (English) Zbl 1278.11053
Summary: In his thesis, J. Weisinger [Some results on classical Eisenstein series and modular forms over function fields. Thesis, Harvard Univ. (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 A. Wiles [Ann. Math. (2) 123, No. 3, 407–456 (1986; Zbl 0613.12013)]. 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 $$\mathcal E_k(\mathcal N, \varPsi)$$ denote the space of Hilbert modular Eisenstein series of parallel weight $$k\geq 3$$, level $$\mathcal{N}$$ and Hecke character $$\varPsi$$ 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 $$\mathcal E_k(\mathcal N, \varPsi)$$ 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 A. O. L. Atkin and W.-C. W. Li [Invent. Math. 48, 221–243 (1978; Zbl 0369.10016)].

##### MSC:
 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F11 Holomorphic modular forms of integral weight
##### Keywords:
Hilbert modular form; Eisenstein series; newform
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##### References:
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