scholarly journals Hecke Operators and Hilbert Modular Forms

2008 ◽  
pp. 387-401 ◽  
Author(s):  
Paul E. Gunnells ◽  
Dan Yasaki
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

scholarly journals HECKE OPERATORS ON HILBERT–SIEGEL MODULAR FORMS

2007 ◽  
Vol 03 (03) ◽  
pp. 391-420 ◽  
Author(s):  
SUZANNE CAULK ◽  
LYNNE H. WALLING
Keyword(s):  
Number Field ◽  
Modular Forms ◽  
Product Space ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Linear Operators ◽  
Hecke Operators ◽  
Hilbert Modular Forms ◽  
Linear Transformations ◽  
Hilbert Modular

We define Hilbert–Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals).

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

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