Overconvergent Eichler-Shimura Isomorphisms for Unitary Shimura Curves over Totally Real Fields

2021 ◽  
Author(s):  
Daniel Barrera ◽  
Shan Gao
Keyword(s):  
Shimura Curves ◽  
Totally Real ◽  
Totally Real Fields

scholarly journals p-adic modular forms of non-integral weight over Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 32-62 ◽  
Author(s):  
Riccardo Brasca
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Hecke Operators ◽  
Analytic Space ◽  
Shimura Curves ◽  
Rigid Analytic Space ◽  
Integral Weight ◽  
Set Up ◽  
Totally Real ◽  
Totally Real Fields

AbstractIn this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

Introduction and Statement of Main Results

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Abelian Varieties ◽  
Main Idea ◽  
Shimura Curves ◽  
Modular Curves ◽  
Heegner Points ◽  
Cm Points ◽  
Totally Real ◽  
Totally Real Fields ◽  
Derivatives Of ◽  
Original Formula

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

scholarly journals Ihara’s Lemma for Shimura curves over totally real fields via patching

2020 ◽  
Author(s):  
Jeffrey Manning ◽  
Jack Shotton
Keyword(s):  
Maximal Ideal ◽  
Hecke Algebra ◽  
Galois Representation ◽  
Shimura Curves ◽  
The Cost ◽  
Totally Real ◽  
Totally Real Fields

Abstract We prove Ihara’s lemma for the mod l cohomology of Shimura curves, localized at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves over $$\mathbb {Q}$$ Q , under various assumptions on l. Our method is totally different and can avoid these assumptions, at the cost of imposing the large image hypothesis. It uses the Taylor–Wiles method, as improved by Diamond and Kisin, and the geometry of integral models of Shimura curves at an auxiliary prime.

scholarly journals Λ-adic modular symbols over totally real fields

2011 ◽  
pp. 841-865 ◽  
Author(s):  
Baskar Balasubramanyam ◽  
Matteo Longo
Keyword(s):  
Modular Symbols ◽  
Totally Real ◽  
Totally Real Fields

scholarly journals Motives over totally real fields and $p$-adic $L$-functions

1994 ◽  
Vol 44 (4) ◽  
pp. 989-1023 ◽  
Author(s):  
Alexei A. Panchishkin
Keyword(s):  
Totally Real ◽  
Totally Real Fields
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