scholarly journals Non-vanishing of derivatives of L-functions of Hilbert modular forms in the critical strip

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Alia Hamieh ◽  
Wissam Raji
Keyword(s):  
Modular Forms ◽  
Hilbert Modular Forms ◽  
Critical Strip ◽  
Hilbert Modular ◽  
Derivatives Of

NON-VANISHING OF DERIVATIVES OF L-FUNCTIONS ATTACHED TO HILBERT MODULAR FORMS

2012 ◽  
Vol 08 (04) ◽  
pp. 1099-1105 ◽  
Author(s):  
NAOMI TANABE
Keyword(s):  
Number Theory ◽  
Modular Forms ◽  
Critical Value ◽  
Transcendental Number ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Derivatives Of

This paper is to show a non-vanishing property of the derivative of certain L-functions. For certain primitive holomorphic Hilbert modular forms, if the central critical value of the standard L-function does not vanish, then neither does its derivative. This is a generalization of a result by Gun, Murty and Rath in the case of elliptic modular forms. Some applications in transcendental number theory deduced from this result are discussed as well.

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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