scholarly journals Siegel modular forms of degree two attached to Hilbert modular forms

2012 ◽  
Vol 132 (4) ◽  
pp. 543-564 ◽  
Author(s):  
Jennifer Johnson-Leung ◽  
Brooks Roberts
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

2020 ◽  
pp. 1-22
Author(s):  
JIM BROWN ◽  
HUIXI LI
Keyword(s):  
Functional Equation ◽  
Modular Forms ◽  
Class Number ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Free Level ◽  
Totally Real

Abstract It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

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.

scholarly journals THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

2015 ◽  
Vol 3 ◽  
Author(s):  
XIN WAN
Keyword(s):  
Modular Forms ◽  
Base Change ◽  
Hilbert Modular Forms ◽  
Selmer Group ◽  
Hilbert Modular Form ◽  
Main Conjecture ◽  
Hilbert Modular ◽  
Trivial Character ◽  
The One ◽  
Ordinary Reduction

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing$p$, if the central critical$L$-value is zero then the$p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

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