scholarly journals Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields

2007 ◽  
Vol 125 (4) ◽  
pp. 427-470 ◽  
Author(s):  
Tobias Berger
Keyword(s):  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Eisenstein Cohomology

scholarly journals On the Eisenstein ideal for imaginary quadratic fields

2009 ◽  
Vol 145 (03) ◽  
pp. 603-632 ◽  
Author(s):  
Tobias Berger
Keyword(s):  
Lower Bound ◽  
Automorphic Forms ◽  
Quadratic Field ◽  
Galois Representations ◽  
Hecke Algebra ◽  
Quadratic Fields ◽  
Selmer Group ◽  
Imaginary Quadratic Fields ◽  
Eisenstein Ideal ◽  
Eisenstein Cohomology

AbstractFor certain algebraic Hecke charactersχof an imaginary quadratic fieldFwe define an Eisenstein ideal in ap-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the specialL-valueL(0,χ). We further prove that its index is bounded from above by thep-valuation of the order of the Selmer group of thep-adic Galois character associated toχ−1. This uses the work of R. Tayloret al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms ofL(0,χ), coinciding with the value given by the Bloch–Kato conjecture.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

Sign in / Sign up

Export Citation Format

Share Document