Newform theory for Hilbert Eisenstein series

作者：Atwill Timothy W; Linowitz Benjamin^*

来源：Ramanujan Journal, 2013, 30(2): 257-278.

DOI：10.1007/s11139-012-9418-2

• Hilbert modular form
• Eisenstein series
• Newform

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 Hilbertmodular 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 1/2. Additionally, we provide a number of applications of this newform theory. Let E-k(N, Psi) denote the space of Hilbert modular Eisenstein series of parallel weight k >= 3, level N and Hecke character Psi over a totally real field K. For any prime q dividing N, we define an operator C-q generalizing the Hecke operator T-q and prove a multiplicity-one theorem for E-k(N,Psi) with respect to the algebra generated by the Hecke operators T-p (p (sic) N) and the operators C-q (q vertical bar 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).

• 出版日期2013-2