scholarly journals On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

2006 ◽  
Vol 56 (3) ◽  
pp. 689-733 ◽  
Author(s):  
Matteo Longo
Keyword(s):  
Elliptic Curves ◽  
Totally Real ◽  
Totally Real Fields

scholarly journals Modularity of nearly ordinary 2-adic residually dihedral Galois representations

2014 ◽  
Vol 150 (8) ◽  
pp. 1235-1346 ◽  
Author(s):  
Patrick B. Allen
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Hida Families ◽  
Totally Real ◽  
Totally Real Fields ◽  
Totally Real Field

AbstractWe prove modularity of some two-dimensional,$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above$2$and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the$2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

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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