scholarly journals Automorphic forms and cohomology theories on Shimura curves of small discriminant

2010 ◽  
Vol 225 (2) ◽  
pp. 1013-1045 ◽  
Author(s):  
Michael Hill ◽  
Tyler Lawson
Keyword(s):  
Automorphic Forms ◽  
Shimura Curves

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 Schwarzian differential equations and Hecke eigenforms on Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Yifan Yang
Keyword(s):  
Differential Equations ◽  
Automorphic Forms ◽  
Hecke Operators ◽  
Shimura Curves ◽  
Genus Zero ◽  
Shimura Curve ◽  
Hecke Eigenforms ◽  
Image Position

AbstractLet X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

scholarly journals Computing fundamental domains for the Bruhat–Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves

2014 ◽  
Vol 17 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Cameron Franc ◽  
Marc Masdeu
Keyword(s):  
High Precision ◽  
Modular Forms ◽  
Automorphic Forms ◽  
Shimura Curves ◽  
Discrete Groups ◽  
Canonical Embedding ◽  
Quaternion Algebras ◽  
Shimura Curve ◽  
Fundamental Domains

AbstractWe describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

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