Newform theory for Hilbert Eisenstein series.

*(English)*Zbl 1278.11053Summary: 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 |

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\textit{T. W. Atwill} and \textit{B. Linowitz}, Ramanujan J. 30, No. 2, 257--278 (2013; Zbl 1278.11053)

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##### References:

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