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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
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[1] Atkin, A.O.L., Lehner, J.: Hecke operators on {\(\Gamma\)} 0(m). Math. Ann. 185, 134–160 (1970) · Zbl 0185.15502
[2] Atkin, A.O.L., Li, W.C.W.: Twists of newforms and pseudo-eigenvalues of W-operators. Invent. Math. 48(3), 221–243 (1978) · Zbl 0377.10017
[3] Atwill, T.W.: Diagonalizing Hilbert cusp forms. Pac. J. Math. 169(1), 33–49 (1995) · Zbl 0858.11026
[4] Darmon, H., Dasgupta, S., Pollack, R.: Hilbert modular forms and the Gross–Stark conjecture. Ann. Math. 174(1), 439–484 (2011) · Zbl 1250.11099
[5] Heilbronn, H.: Zeta-functions and L-functions. In: Algebraic Number Theory. Proc. Instructional Conf., Brighton, 1965, pp. 204–230. Thompson, Washington (1967)
[6] Hijikata, H., Pizer, A.K., Shemanske, T.R.: Twists of newforms. J. Number Theory 35(3), 287–324 (1990) · Zbl 0714.11028
[7] Kitaoka, Y.: A note on Hecke operators and theta-series. Nagoya Math. J. 42, 189–195 (1971) · Zbl 0221.10031
[8] Li, W.C.W.: Newforms and functional equations. Math. Ann. 212, 285–315 (1975) · Zbl 0286.10016
[9] Li, W.-C.W.: Diagonalizing modular forms. J. Algebra 99(1), 210–231 (1986) · Zbl 0594.10018
[10] Linowitz, B.: Decomposition theorems for Hilbert modular newforms. Funct. Approx. Comment. Math. (2012, to appear) · Zbl 1278.11054
[11] Pizer, A.: Hecke operators for {\(\Gamma\)} 0(N). J. Algebra 83(1), 39–64 (1983) · Zbl 0515.10020
[12] Ramakrishnan, D.: A refinement of the strong multiplicity one theorem for GL(2). Appendix to: ”l-adic representations associated to modular forms over imaginary quadratic fields. II” [Invent. Math. 116(1–3), 619–643 (1994), MR1253207 (95h:11050a)] by R. Taylor. Invent. Math. 116(1–3), 645–649 (1994) · Zbl 0823.11021
[13] Shahidi, F.: Best estimates for Fourier coefficients of Maass forms. In: Automorphic forms and analytic number theory, Montreal, PQ, 1989, pp. 135–141. Univ. MontrĂ©al, Montreal (1990) · Zbl 0748.11025
[14] Shemanske, T.R., Walling, L.H.: Twists of Hilbert modular forms. Trans. Am. Math. Soc. 338(1), 375–403 (1993) · Zbl 0785.11029
[15] Shimura, G.: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups. Ann. Math. (2) 111(2), 313–375 (1980) · Zbl 0438.12003
[16] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679 (1978) · Zbl 0394.10015
[17] Weisinger, J.: Some results on classical Eisenstein series and modular forms over function fields. Thesis, Harvard Univ. (1977)
[18] Wiles, A.: On p-adic representations for totally real fields. Ann. Math. (2) 123(3), 407–456 (1986) · Zbl 0613.12013
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