scholarly journals Hilbert modular forms and the Ramanujan conjecture

2007 ◽  
pp. 35-56 ◽  
Author(s):  
Don Blasius
Keyword(s):  
Modular Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Ramanujan Conjecture

scholarly journals On the equality case of the Ramanujan Conjecture for Hilbert modular forms

2019 ◽  
Vol 15 (10) ◽  
pp. 2107-2114
Author(s):  
Liubomir Chiriac
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Complex Multiplication ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Automorphic Representations ◽  
Equality Case ◽  
Hilbert Modular ◽  
Ramanujan Conjecture ◽  
Satake Parameters

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

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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